{"id":852,"date":"2025-06-20T17:17:46","date_gmt":"2025-06-20T17:17:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=852"},"modified":"2025-08-28T13:20:06","modified_gmt":"2025-08-28T13:20:06","slug":"sequences-and-series-foundations-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/sequences-and-series-foundations-background-youll-need-3\/","title":{"raw":"Sequences and Series Foundations: Background You'll Need 3","rendered":"Sequences and Series Foundations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Determine limits by applying the squeeze theorem<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The Squeeze Theorem<\/h2>\r\n<p id=\"fs-id1170572611898\">When evaluating limits of trigonometric functions, we need a specialized approach since standard algebraic techniques may not apply directly. The <strong>squeeze theorem<\/strong> proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by \"squeezing\" a function, with a limit at a point [latex]a[\/latex] that is unknown, between two functions having a common known limit at [latex]a[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203431\/CNX_Calc_Figure_02_03_005.jpg\" alt=\"A graph of three functions over a small interval. All three functions curve. Over this interval, the function g(x) is trapped between the functions h(x), which gives greater y values for the same x values, and f(x), which gives smaller y values for the same x values. The functions all approach the same limit when x=a.\" width=\"487\" height=\"462\" \/> Figure 5. The Squeeze Theorem applies when [latex]f(x)\\le g(x)\\le h(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim}f(x)=\\underset{x\\to a}{\\lim}h(x)[\/latex].[\/caption]<section class=\"textbox keyTakeaway\">\r\n<h3>the squeeze theorem<\/h3>\r\n<p id=\"fs-id1170571603686\">Let [latex]f(x), \\, g(x)[\/latex], and [latex]h(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over an open interval containing [latex]a[\/latex]. If<\/p>\r\n\r\n<div id=\"fs-id1170571603742\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g(x)\\le f(x)\\le h(x)[\/latex]<\/div>\r\n<p id=\"fs-id1170571603783\">for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex] and<\/p>\r\n\r\n<div id=\"fs-id1170571603801\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}g(x)=L=\\underset{x\\to a}{\\lim}h(x)[\/latex]<\/div>\r\n<p id=\"fs-id1170571654186\">where [latex]L[\/latex] is a real number, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox questionHelp\"><strong>How To: Solve Trigonometric Limits Using the Squeeze Theorem<\/strong>\r\n<ol>\r\n \t<li>Confirm the function shows an indeterminate form that the Squeeze Theorem can address.<\/li>\r\n \t<li>Find two bounding functions, [latex]g(x)[\/latex] and [latex]h(x)[\/latex], that satisfy [latex]g(x)\\le f(x)\\le h(x)[\/latex].<\/li>\r\n \t<li>Ensure [latex]g(x)[\/latex] and [latex]h(x)[\/latex] approach the same limit at the point of interest.<\/li>\r\n \t<li>Use the established bounds to deduce the limit of [latex]f(x)[\/latex]. If [latex]g(x)[\/latex] and [latex]h(x)[\/latex] have a common limit [latex]L[\/latex], then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Apply the Squeeze Theorem to evaluate the limit [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}[\/latex].First start by identify bounding functions. We know that [latex]-1\\le \\sin{x^2}\\le 1[\/latex].Next, divide these inequalities by [latex]x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\frac{-1}{x}\\le \\sin{x^2}\\le \\frac{1}{x}[\/latex]<\/p>\r\nNow, evaluate the limits of the bounding functions as [latex]x[\/latex] approaches [latex]0[\/latex].\r\n\r\nBoth [latex]\\frac{-1}{x}[\/latex] and [latex]\\frac{1}{x}[\/latex] approach infinity as [latex]x[\/latex] approaches [latex]0[\/latex], but since [latex]\\frac{\\sin{x^2}}{x}[\/latex]\u00a0is sandwiched between them, we deduce that\r\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}=0[\/latex]<\/p>\r\ndue to the squeeze theorem.\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170571654238\">Apply the Squeeze Theorem to evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571654269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571654269\"]\r\n\r\nTo evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex]\u00a0using the Squeeze Theorem:\r\n<ol>\r\n \t<li>Recognize that [latex]-1\\le \\cos x\\le 1[\/latex] for all real numbers.<\/li>\r\n \t<li>Multiply this inequality by [latex]x[\/latex] to get [latex]-|x|\\le x \\cos x\\le |x|[\/latex]<\/li>\r\n \t<li>As [latex]x[\/latex] approaches [latex]0[\/latex], both [latex]\u2212\u2223x\u2223[\/latex] and [latex]\u2223x\u2223[\/latex] approach [latex]0[\/latex].<\/li>\r\n \t<li>By the Squeeze Theorem, since [latex]x\\cos{x}[\/latex] is squeezed between two functions that both approach [latex]0[\/latex], [latex]\\underset{x\\to 0}{\\lim}x \\cos x=0[\/latex]. The graphs of [latex]f(x)=-|x|, \\, g(x)=x \\cos x[\/latex], and [latex]h(x)=|x|[\/latex] are shown in Figure 6.<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_1644\" align=\"alignnone\" width=\"309\"]<img class=\"wp-image-1644 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2023\/09\/19035449\/Screenshot-2024-04-18-235434.png\" alt=\"The graph of three functions: h(x) = x, f(x) = -x, and g(x) = xcos(x). The first, h(x) = x, is a linear function with slope of 1 going through the origin. The second, f(x), is also a linear function with slope of \u22121; going through the origin. The third, g(x) = xcos(x), curves between the two and goes through the origin. It opens upward for x&gt;0 and downward for x&gt;0.\" width=\"309\" height=\"293\" \/> Figure 6. The graphs of \ud835\udc53(\ud835\udc65), \ud835\udc54(\ud835\udc65), and \u210e(\ud835\udc65) are shown around the point \ud835\udc65=0.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_number=1]204232[\/ohm_question]<\/section>We now use the squeeze theorem to tackle several very important limits. The first of these limits is [latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta[\/latex].\r\n<h3>Evaluating the Limit of Sine as Theta Approaches Zero<\/h3>\r\nConsider the behavior of [latex]\\sin(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero. On the unit circle, [latex]\\sin(\\theta)[\/latex]\u00a0corresponds to the [latex]y[\/latex]-coordinate, which also represents the arc's height for a given angle, [latex]\\theta[\/latex].\r\n\r\nAs [latex]\\theta[\/latex] gets closer to zero, particularly for [latex]0 &lt; \\theta &lt; \\frac{\\pi}{2}[\/latex], [latex]\\sin(\\theta)[\/latex] becomes smaller and approaches the angle's measure itself, meaning [latex]\\sin(\\theta)[\/latex] is squeezed between [latex]0[\/latex] and [latex]\\theta[\/latex] .\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203438\/CNX_Calc_Figure_02_03_007.jpg\" alt=\"A diagram of the unit circle in the x,y plane \u2013 it is a circle with radius 1 and center at the origin. A specific point (cos(theta), sin(theta)) is labeled in quadrant 1 on the edge of the circle. This point is one vertex of a right triangle inside the circle, with other vertices at the origin and (cos(theta), 0). As such, the lengths of the sides are cos(theta) for the base and sin(theta) for the height, where theta is the angle created by the hypotenuse and base. The radian measure of angle theta is the length of the arc it subtends on the unit circle. The diagram shows that for 0 &lt; theta &lt; pi\/2, 0 &lt; sin(theta) &lt; theta.\" width=\"487\" height=\"425\" \/> Figure 7. The sine function is shown as a line on the unit circle.[\/caption]\r\n\r\nFirst, consider the established inequalities for [latex]\\sin(\\theta)[\/latex] when [latex]\\theta[\/latex] \u00a0is between [latex] 0[\/latex] and [latex]\\frac{\\pi}{2}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]0 &lt; \\theta &lt; \\frac{\\pi}{2} \\Longrightarrow 0 &lt; \\sin \\theta &lt; \\theta[\/latex]<\/p>\r\nNow, as [latex]\\theta[\/latex] approaches zero from the positive direction, we know that [latex]\\sin(\\theta)[\/latex]\u00a0also approaches zero because it is sandwiched between [latex]0[\/latex] and [latex]\\theta[\/latex].\r\n\r\nMathematically, this can be expressed as:\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^+}{\\lim}0=0 \\text{\u00a0 \u00a0 \u00a0and\u00a0 \u00a0 \u00a0}\\underset{\\theta \\to 0^+}{\\lim} \\theta =0[\/latex],<\/p>\r\nwhich, according to the Squeeze Theorem, compels [latex]\\sin(\\theta)[\/latex] to satisfy:\r\n<div id=\"fs-id1170571545491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^+}{\\lim} \\sin \\theta =0[\/latex].<\/div>\r\nThe same principle applies when approaching zero from the negative side, where [latex]\\sin(\\theta)[\/latex] is negative but greater than [latex]-\\theta[\/latex]:\r\n<div style=\"text-align: center;\">[latex]\\frac{-\\pi}{2} &lt; \\theta &lt; 0 \\Longrightarrow-\\theta &lt; \\sin \\theta &lt; 0[\/latex]<\/div>\r\n<div>Here too, as [latex]\\theta[\/latex] approaches zero, [latex]\\sin(\\theta)[\/latex] \u00a0is \"squeezed\" to zero:<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^-}{\\lim}0=0 \\text{\u00a0 \u00a0 \u00a0and\u00a0 \u00a0 \u00a0}\\underset{\\theta \\to 0^-}{\\lim} (-\\theta) =0[\/latex],<\/div>\r\n<div>leading to the conclusion that:<\/div>\r\n<div><\/div>\r\n<div style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^-}{\\lim} \\sin \\theta =0[\/latex].<\/div>\r\n<div>Therefore, we can definitively state that the limit of [latex]\\sin(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero from either direction is [latex]0[\/latex].<\/div>\r\n<div><section class=\"textbox keyTakeaway\">\r\n<h3>the limit of [latex]\\sin(\\theta)[\/latex]<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex]<\/p>\r\n\r\n<\/section><\/div>\r\n<h3>Evaluating the Limit of Cosine as Theta Approaches Zero<\/h3>\r\n<div>\r\n\r\nTo evaluate the limit of [latex]\\cos(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero, we rely on the fundamental Pythagorean identity which states that for any angle [latex]\\theta[\/latex], the square of the cosine of [latex]\\theta[\/latex] plus the square of the sine of [latex]\\theta[\/latex]\u00a0equals one:\r\n<p style=\"text-align: center;\">[latex]\\cos^2(\\theta)+\\sin^2(\\theta)=1[\/latex]<\/p>\r\nRearranging this identity, we can isolate [latex]\\cos(\\theta)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\cos(\\theta)=\\sqrt{1\u2212\\sin^2(\\theta)}[\/latex]<\/p>\r\nSince the sine function is bounded between [latex]-1[\/latex] and [latex]1 [\/latex] for all [latex]\\theta[\/latex], and as [latex]\\theta[\/latex] approaches zero, [latex]\\sin(\\theta)[\/latex]\u00a0also approaches zero, we can substitute this limit into our identity:\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta=\\underset{\\theta \\to 0}{\\lim} \\sqrt{1\u2212\\sin^2(\\theta)} [\/latex]<\/p>\r\nGiven that [latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex], we then have:\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\sqrt{1\u2212\\sin^2(\\theta)} =\\sqrt{1-0^2}=1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Thus, we confirm that the limit of [latex]\\cos(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero is [latex]1[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the limit of [latex]\\cos(\\theta)[\/latex]<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta =1[\/latex]<\/p>\r\n\r\n<\/section><\/div>\r\n<h3>Exploring the Limit of Sine Theta Over Theta<\/h3>\r\n<div>A pivotal limit in calculus, particularly relevant in the study of derivatives and integrals of trigonometric functions, is [latex]\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta}[\/latex].<\/div>\r\n<div><\/div>\r\n<div>To understand this limit, we look to the unit circle, where the sine and tangent functions provide geometric insights into this foundational limit.<\/div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203441\/CNX_Calc_Figure_02_03_008.jpg\" alt=\"The same diagram as the previous one. However, the triangle is expanded. The base is now from the origin to (1,0). The height goes from (1,0) to (1, tan(theta)). The hypotenuse goes from the origin to (1, tan(theta)). As such, the height is now tan(theta). It shows that for 0 &lt; theta &lt; pi\/2, sin(theta) &lt; theta &lt; tan(theta).\" width=\"487\" height=\"478\" \/> Figure 8. The sine and tangent functions are shown as lines on the unit circle.[\/caption]\r\n\r\nAnalyze the behavior of [latex]\\sin(\\theta)[\/latex] and [latex]\\tan(\\theta)[\/latex] within the first quadrant of the unit circle, specifically for angles [latex]\\theta[\/latex] where [latex]0 &lt; \\theta &lt; \\frac{\\pi}{2} [\/latex].\r\n\r\nIn this range, it's clear from the geometric representation that [latex]\\sin(\\theta)[\/latex]\u00a0is always less than the length of the tangent line segment from the point on the circle to the [latex]x[\/latex]-axis, which is [latex]\\tan(\\theta)[\/latex]. Consequently, we have the inequality:\r\n\r\n<\/div>\r\n<div style=\"text-align: center;\">[latex]0&lt; \\sin \\theta &lt; \\tan \\theta[\/latex]<\/div>\r\n<p id=\"fs-id1170571649306\">By dividing each term in the inequality by [latex]\\sin \\theta [\/latex] , we are led to:<\/p>\r\n\r\n<div id=\"fs-id1170571649320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1 &lt; \\dfrac{\\theta}{\\sin \\theta} &lt; \\dfrac{1}{\\cos \\theta}[\/latex]<\/div>\r\nWith the reciprocal, this inequality can be restated as:\r\n<div id=\"fs-id1170571649362\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1 &gt; \\dfrac{\\sin \\theta}{\\theta} &gt; \\cos \\theta[\/latex]<\/div>\r\nAs [latex]\\theta[\/latex] approaches zero, [latex]\\cos(\\theta)[\/latex]\u00a0approaches [latex]1[\/latex]. Therefore, [latex]\\sin(\\theta)[\/latex] is squeezed between [latex]\\cos(\\theta)[\/latex]\u00a0and [latex]1[\/latex].\r\n\r\nSince [latex]\\cos(\\theta)[\/latex]\u00a0also approaches [latex]1[\/latex] as [latex]\\theta[\/latex] \u00a0approaches zero, the Squeeze Theorem can be applied to conclude that:\r\n<div id=\"fs-id1170571611730\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{\\sin \\theta}{\\theta}=1[\/latex]<\/div>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>the limit of [latex]\\dfrac{\\sin \\theta}{\\theta}[\/latex]<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{\\sin \\theta}{\\theta}=1[\/latex]<\/p>\r\n\r\n<\/section>\r\n<h3>Evaluating the Limit of [latex]\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/h3>\r\nAs we build upon the understanding of limits involving trigonometric functions, the next step is to apply the Squeeze Theorem to evaluate limits that are not immediately obvious.\r\n<p id=\"fs-id1170571611766\">In the example below, we use the limit of [latex]\\frac{\\sin {\\theta}}{\\theta}[\/latex] to establish [latex]\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}=0[\/latex]. This limit also proves useful in later modules.<\/p>\r\n\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572243724\">Evaluate [latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572243764\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572243764\"]\r\n<p id=\"fs-id1170572243764\">In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine:<\/p>\r\n\r\n<div id=\"fs-id1170572243769\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc} \\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}&amp; =\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta} \\cdot \\frac{1+ \\cos \\theta}{1+ \\cos \\theta} \\\\ &amp; =\\underset{\\theta \\to 0}{\\lim}\\frac{1-\\cos^2 \\theta}{\\theta(1+ \\cos \\theta)} \\\\ &amp; =\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin^2 \\theta}{\\theta(1+ \\cos \\theta)} \\\\ &amp; =\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta} \\cdot \\frac{\\sin \\theta}{1+ \\cos \\theta} \\\\ &amp; =1 \\cdot \\frac{0}{2}=0 \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170571652241\">Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170571652244\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}=0[\/latex]\r\n[\/hidden-answer]<\/div>\r\n<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>the limit of [latex]\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}=0[\/latex]<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Determine limits by applying the squeeze theorem<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>The Squeeze Theorem<\/h2>\n<p id=\"fs-id1170572611898\">When evaluating limits of trigonometric functions, we need a specialized approach since standard algebraic techniques may not apply directly. The <strong>squeeze theorem<\/strong> proves very useful for establishing basic trigonometric limits. This theorem allows us to calculate limits by &#8220;squeezing&#8221; a function, with a limit at a point [latex]a[\/latex] that is unknown, between two functions having a common known limit at [latex]a[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203431\/CNX_Calc_Figure_02_03_005.jpg\" alt=\"A graph of three functions over a small interval. All three functions curve. Over this interval, the function g(x) is trapped between the functions h(x), which gives greater y values for the same x values, and f(x), which gives smaller y values for the same x values. The functions all approach the same limit when x=a.\" width=\"487\" height=\"462\" \/><figcaption class=\"wp-caption-text\">Figure 5. The Squeeze Theorem applies when [latex]f(x)\\le g(x)\\le h(x)[\/latex] and [latex]\\underset{x\\to a}{\\lim}f(x)=\\underset{x\\to a}{\\lim}h(x)[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3>the squeeze theorem<\/h3>\n<p id=\"fs-id1170571603686\">Let [latex]f(x), \\, g(x)[\/latex], and [latex]h(x)[\/latex] be defined for all [latex]x\\ne a[\/latex] over an open interval containing [latex]a[\/latex]. If<\/p>\n<div id=\"fs-id1170571603742\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]g(x)\\le f(x)\\le h(x)[\/latex]<\/div>\n<p id=\"fs-id1170571603783\">for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex] and<\/p>\n<div id=\"fs-id1170571603801\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to a}{\\lim}g(x)=L=\\underset{x\\to a}{\\lim}h(x)[\/latex]<\/div>\n<p id=\"fs-id1170571654186\">where [latex]L[\/latex] is a real number, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How To: Solve Trigonometric Limits Using the Squeeze Theorem<\/strong><\/p>\n<ol>\n<li>Confirm the function shows an indeterminate form that the Squeeze Theorem can address.<\/li>\n<li>Find two bounding functions, [latex]g(x)[\/latex] and [latex]h(x)[\/latex], that satisfy [latex]g(x)\\le f(x)\\le h(x)[\/latex].<\/li>\n<li>Ensure [latex]g(x)[\/latex] and [latex]h(x)[\/latex] approach the same limit at the point of interest.<\/li>\n<li>Use the established bounds to deduce the limit of [latex]f(x)[\/latex]. If [latex]g(x)[\/latex] and [latex]h(x)[\/latex] have a common limit [latex]L[\/latex], then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Apply the Squeeze Theorem to evaluate the limit [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}[\/latex].First start by identify bounding functions. We know that [latex]-1\\le \\sin{x^2}\\le 1[\/latex].Next, divide these inequalities by [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{-1}{x}\\le \\sin{x^2}\\le \\frac{1}{x}[\/latex]<\/p>\n<p>Now, evaluate the limits of the bounding functions as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\n<p>Both [latex]\\frac{-1}{x}[\/latex] and [latex]\\frac{1}{x}[\/latex] approach infinity as [latex]x[\/latex] approaches [latex]0[\/latex], but since [latex]\\frac{\\sin{x^2}}{x}[\/latex]\u00a0is sandwiched between them, we deduce that<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}=0[\/latex]<\/p>\n<p>due to the squeeze theorem.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170571654238\">Apply the Squeeze Theorem to evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571654269\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571654269\" class=\"hidden-answer\" style=\"display: none\">\n<p>To evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex]\u00a0using the Squeeze Theorem:<\/p>\n<ol>\n<li>Recognize that [latex]-1\\le \\cos x\\le 1[\/latex] for all real numbers.<\/li>\n<li>Multiply this inequality by [latex]x[\/latex] to get [latex]-|x|\\le x \\cos x\\le |x|[\/latex]<\/li>\n<li>As [latex]x[\/latex] approaches [latex]0[\/latex], both [latex]\u2212\u2223x\u2223[\/latex] and [latex]\u2223x\u2223[\/latex] approach [latex]0[\/latex].<\/li>\n<li>By the Squeeze Theorem, since [latex]x\\cos{x}[\/latex] is squeezed between two functions that both approach [latex]0[\/latex], [latex]\\underset{x\\to 0}{\\lim}x \\cos x=0[\/latex]. The graphs of [latex]f(x)=-|x|, \\, g(x)=x \\cos x[\/latex], and [latex]h(x)=|x|[\/latex] are shown in Figure 6.<\/li>\n<\/ol>\n<figure id=\"attachment_1644\" aria-describedby=\"caption-attachment-1644\" style=\"width: 309px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1644 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2023\/09\/19035449\/Screenshot-2024-04-18-235434.png\" alt=\"The graph of three functions: h(x) = x, f(x) = -x, and g(x) = xcos(x). The first, h(x) = x, is a linear function with slope of 1 going through the origin. The second, f(x), is also a linear function with slope of \u22121; going through the origin. The third, g(x) = xcos(x), curves between the two and goes through the origin. It opens upward for x&gt;0 and downward for x&gt;0.\" width=\"309\" height=\"293\" \/><figcaption id=\"caption-attachment-1644\" class=\"wp-caption-text\">Figure 6. The graphs of \ud835\udc53(\ud835\udc65), \ud835\udc54(\ud835\udc65), and \u210e(\ud835\udc65) are shown around the point \ud835\udc65=0.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm204232\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=204232&theme=lumen&iframe_resize_id=ohm204232&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We now use the squeeze theorem to tackle several very important limits. The first of these limits is [latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta[\/latex].<\/p>\n<h3>Evaluating the Limit of Sine as Theta Approaches Zero<\/h3>\n<p>Consider the behavior of [latex]\\sin(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero. On the unit circle, [latex]\\sin(\\theta)[\/latex]\u00a0corresponds to the [latex]y[\/latex]-coordinate, which also represents the arc&#8217;s height for a given angle, [latex]\\theta[\/latex].<\/p>\n<p>As [latex]\\theta[\/latex] gets closer to zero, particularly for [latex]0 < \\theta < \\frac{\\pi}{2}[\/latex], [latex]\\sin(\\theta)[\/latex] becomes smaller and approaches the angle&#8217;s measure itself, meaning [latex]\\sin(\\theta)[\/latex] is squeezed between [latex]0[\/latex] and [latex]\\theta[\/latex] .\n\n\n\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203438\/CNX_Calc_Figure_02_03_007.jpg\" alt=\"A diagram of the unit circle in the x,y plane \u2013 it is a circle with radius 1 and center at the origin. A specific point (cos(theta), sin(theta)) is labeled in quadrant 1 on the edge of the circle. This point is one vertex of a right triangle inside the circle, with other vertices at the origin and (cos(theta), 0). As such, the lengths of the sides are cos(theta) for the base and sin(theta) for the height, where theta is the angle created by the hypotenuse and base. The radian measure of angle theta is the length of the arc it subtends on the unit circle. The diagram shows that for 0 &lt; theta &lt; pi\/2, 0 &lt; sin(theta) &lt; theta.\" width=\"487\" height=\"425\" \/><figcaption class=\"wp-caption-text\">Figure 7. The sine function is shown as a line on the unit circle.<\/figcaption><\/figure>\n<p>First, consider the established inequalities for [latex]\\sin(\\theta)[\/latex] when [latex]\\theta[\/latex] \u00a0is between [latex]0[\/latex] and [latex]\\frac{\\pi}{2}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]0 < \\theta < \\frac{\\pi}{2} \\Longrightarrow 0 < \\sin \\theta < \\theta[\/latex]<\/p>\n<p>Now, as [latex]\\theta[\/latex] approaches zero from the positive direction, we know that [latex]\\sin(\\theta)[\/latex]\u00a0also approaches zero because it is sandwiched between [latex]0[\/latex] and [latex]\\theta[\/latex].<\/p>\n<p>Mathematically, this can be expressed as:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^+}{\\lim}0=0 \\text{\u00a0 \u00a0 \u00a0and\u00a0 \u00a0 \u00a0}\\underset{\\theta \\to 0^+}{\\lim} \\theta =0[\/latex],<\/p>\n<p>which, according to the Squeeze Theorem, compels [latex]\\sin(\\theta)[\/latex] to satisfy:<\/p>\n<div id=\"fs-id1170571545491\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^+}{\\lim} \\sin \\theta =0[\/latex].<\/div>\n<p>The same principle applies when approaching zero from the negative side, where [latex]\\sin(\\theta)[\/latex] is negative but greater than [latex]-\\theta[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{-\\pi}{2} < \\theta < 0 \\Longrightarrow-\\theta < \\sin \\theta < 0[\/latex]<\/div>\n<div>Here too, as [latex]\\theta[\/latex] approaches zero, [latex]\\sin(\\theta)[\/latex] \u00a0is &#8220;squeezed&#8221; to zero:<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^-}{\\lim}0=0 \\text{\u00a0 \u00a0 \u00a0and\u00a0 \u00a0 \u00a0}\\underset{\\theta \\to 0^-}{\\lim} (-\\theta) =0[\/latex],<\/div>\n<div>leading to the conclusion that:<\/div>\n<div><\/div>\n<div style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0^-}{\\lim} \\sin \\theta =0[\/latex].<\/div>\n<div>Therefore, we can definitively state that the limit of [latex]\\sin(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero from either direction is [latex]0[\/latex].<\/div>\n<div>\n<section class=\"textbox keyTakeaway\">\n<h3>the limit of [latex]\\sin(\\theta)[\/latex]<\/h3>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex]<\/p>\n<\/section>\n<\/div>\n<h3>Evaluating the Limit of Cosine as Theta Approaches Zero<\/h3>\n<div>\n<p>To evaluate the limit of [latex]\\cos(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero, we rely on the fundamental Pythagorean identity which states that for any angle [latex]\\theta[\/latex], the square of the cosine of [latex]\\theta[\/latex] plus the square of the sine of [latex]\\theta[\/latex]\u00a0equals one:<\/p>\n<p style=\"text-align: center;\">[latex]\\cos^2(\\theta)+\\sin^2(\\theta)=1[\/latex]<\/p>\n<p>Rearranging this identity, we can isolate [latex]\\cos(\\theta)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\cos(\\theta)=\\sqrt{1\u2212\\sin^2(\\theta)}[\/latex]<\/p>\n<p>Since the sine function is bounded between [latex]-1[\/latex] and [latex]1[\/latex] for all [latex]\\theta[\/latex], and as [latex]\\theta[\/latex] approaches zero, [latex]\\sin(\\theta)[\/latex]\u00a0also approaches zero, we can substitute this limit into our identity:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta=\\underset{\\theta \\to 0}{\\lim} \\sqrt{1\u2212\\sin^2(\\theta)}[\/latex]<\/p>\n<p>Given that [latex]\\underset{\\theta \\to 0}{\\lim} \\sin \\theta =0[\/latex], we then have:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\sqrt{1\u2212\\sin^2(\\theta)} =\\sqrt{1-0^2}=1[\/latex]<\/p>\n<p style=\"text-align: left;\">Thus, we confirm that the limit of [latex]\\cos(\\theta)[\/latex] as [latex]\\theta[\/latex] approaches zero is [latex]1[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>the limit of [latex]\\cos(\\theta)[\/latex]<\/h3>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim} \\cos \\theta =1[\/latex]<\/p>\n<\/section>\n<\/div>\n<h3>Exploring the Limit of Sine Theta Over Theta<\/h3>\n<div>A pivotal limit in calculus, particularly relevant in the study of derivatives and integrals of trigonometric functions, is [latex]\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta}[\/latex].<\/div>\n<div><\/div>\n<div>To understand this limit, we look to the unit circle, where the sine and tangent functions provide geometric insights into this foundational limit.<\/div>\n<div>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203441\/CNX_Calc_Figure_02_03_008.jpg\" alt=\"The same diagram as the previous one. However, the triangle is expanded. The base is now from the origin to (1,0). The height goes from (1,0) to (1, tan(theta)). The hypotenuse goes from the origin to (1, tan(theta)). As such, the height is now tan(theta). It shows that for 0 &lt; theta &lt; pi\/2, sin(theta) &lt; theta &lt; tan(theta).\" width=\"487\" height=\"478\" \/><figcaption class=\"wp-caption-text\">Figure 8. The sine and tangent functions are shown as lines on the unit circle.<\/figcaption><\/figure>\n<p>Analyze the behavior of [latex]\\sin(\\theta)[\/latex] and [latex]\\tan(\\theta)[\/latex] within the first quadrant of the unit circle, specifically for angles [latex]\\theta[\/latex] where [latex]0 < \\theta < \\frac{\\pi}{2}[\/latex].\n\nIn this range, it&#8217;s clear from the geometric representation that [latex]\\sin(\\theta)[\/latex]\u00a0is always less than the length of the tangent line segment from the point on the circle to the [latex]x[\/latex]-axis, which is [latex]\\tan(\\theta)[\/latex]. Consequently, we have the inequality:\n\n<\/div>\n<div style=\"text-align: center;\">[latex]0< \\sin \\theta < \\tan \\theta[\/latex]<\/div>\n<p id=\"fs-id1170571649306\">By dividing each term in the inequality by [latex]\\sin \\theta[\/latex] , we are led to:<\/p>\n<div id=\"fs-id1170571649320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1 < \\dfrac{\\theta}{\\sin \\theta} < \\dfrac{1}{\\cos \\theta}[\/latex]<\/div>\n<p>With the reciprocal, this inequality can be restated as:<\/p>\n<div id=\"fs-id1170571649362\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1 > \\dfrac{\\sin \\theta}{\\theta} > \\cos \\theta[\/latex]<\/div>\n<p>As [latex]\\theta[\/latex] approaches zero, [latex]\\cos(\\theta)[\/latex]\u00a0approaches [latex]1[\/latex]. Therefore, [latex]\\sin(\\theta)[\/latex] is squeezed between [latex]\\cos(\\theta)[\/latex]\u00a0and [latex]1[\/latex].<\/p>\n<p>Since [latex]\\cos(\\theta)[\/latex]\u00a0also approaches [latex]1[\/latex] as [latex]\\theta[\/latex] \u00a0approaches zero, the Squeeze Theorem can be applied to conclude that:<\/p>\n<div id=\"fs-id1170571611730\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{\\sin \\theta}{\\theta}=1[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3>the limit of [latex]\\dfrac{\\sin \\theta}{\\theta}[\/latex]<\/h3>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{\\sin \\theta}{\\theta}=1[\/latex]<\/p>\n<\/section>\n<h3>Evaluating the Limit of [latex]\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/h3>\n<p>As we build upon the understanding of limits involving trigonometric functions, the next step is to apply the Squeeze Theorem to evaluate limits that are not immediately obvious.<\/p>\n<p id=\"fs-id1170571611766\">In the example below, we use the limit of [latex]\\frac{\\sin {\\theta}}{\\theta}[\/latex] to establish [latex]\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}=0[\/latex]. This limit also proves useful in later modules.<\/p>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572243724\">Evaluate [latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572243764\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572243764\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572243764\">In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine:<\/p>\n<div id=\"fs-id1170572243769\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc} \\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta}& =\\underset{\\theta \\to 0}{\\lim}\\frac{1- \\cos \\theta}{\\theta} \\cdot \\frac{1+ \\cos \\theta}{1+ \\cos \\theta} \\\\ & =\\underset{\\theta \\to 0}{\\lim}\\frac{1-\\cos^2 \\theta}{\\theta(1+ \\cos \\theta)} \\\\ & =\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin^2 \\theta}{\\theta(1+ \\cos \\theta)} \\\\ & =\\underset{\\theta \\to 0}{\\lim}\\frac{\\sin \\theta}{\\theta} \\cdot \\frac{\\sin \\theta}{1+ \\cos \\theta} \\\\ & =1 \\cdot \\frac{0}{2}=0 \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170571652241\">Therefore,<\/p>\n<div id=\"fs-id1170571652244\" class=\"equation\" style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}=0[\/latex]\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>the limit of [latex]\\dfrac{1- \\cos \\theta}{\\theta}[\/latex]<\/h3>\n<p style=\"text-align: center;\">[latex]\\underset{\\theta \\to 0}{\\lim}\\dfrac{1- \\cos \\theta}{\\theta}=0[\/latex]<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":671,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/852"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/852\/revisions"}],"predecessor-version":[{"id":2051,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/852\/revisions\/2051"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/671"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/852\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=852"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=852"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=852"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}