{"id":199,"date":"2023-09-20T22:48:13","date_gmt":"2023-09-20T22:48:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-squeeze-theorem\/"},"modified":"2024-08-05T12:34:07","modified_gmt":"2024-08-05T12:34:07","slug":"the-limit-laws-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-limit-laws-learn-it-3\/","title":{"raw":"The Limit Laws: Learn It 3","rendered":"The Limit Laws: Learn It 3"},"content":{"raw":"

The Squeeze Theorem<\/h2>\r\n

The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions.<\/p>\r\n

The next theorem, called the squeeze theorem<\/strong>, proves very useful for establishing basic trigonometric limits.<\/p>\r\n

This theorem allows us to calculate limits by \u201csqueezing\u201d 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].\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A 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]\r\n\r\n

\r\n

The Squeeze Theorem<\/h3>\r\n

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

[latex]g(x)\\le f(x)\\le h(x)[\/latex]<\/div>\r\n

for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex] and<\/p>\r\n

[latex]\\underset{x\\to a}{\\lim}g(x)=L=\\underset{x\\to a}{\\lim}h(x)[\/latex]<\/div>\r\n

where [latex]L[\/latex] is a real number, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/p>\r\n<\/section>\r\n

\r\n

How To: Solve Trigonometric Limits Using the Squeeze Theorem<\/strong><\/p>\r\n

    \r\n\t
  1. Confirm the function shows an indeterminate form that the Squeeze Theorem can address.<\/li>\r\n\t
  2. 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
  3. Ensure [latex]g(x)[\/latex] and [latex]h(x)[\/latex] approach the same limit at the point of interest.<\/li>\r\n\t
  4. 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>\r\n
    \r\n

    Apply the Squeeze Theorem to evaluate the limit [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}[\/latex].<\/p>\r\n

    First start by identify bounding functions. We know that [latex]-1\\le \\sin{x^2}\\le 1[\/latex].
    \r\n
    \r\nNext, divide these inequalities by [latex]x[\/latex].<\/p>\r\n

    [latex]\\frac{-1}{x}\\le \\sin{x^2}\\le \\frac{1}{x}[\/latex]<\/p>\r\n

    Now, evaluate the limits of the bounding functions as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\r\n

    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>\r\n

    [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}=0[\/latex]<\/p>\r\n

    due to the squeeze theorem.<\/p>\r\n<\/section>\r\n

    \r\n

    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\"]<\/p>\r\n

    To evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex]\u00a0using the Squeeze Theorem:<\/p>\r\n

      \r\n\t
    1. Recognize that [latex]-1\\le \\cos x\\le 1[\/latex] for all real numbers.<\/li>\r\n\t
    2. Multiply this inequality by [latex]x[\/latex] to get [latex]-|x|\\le x \\cos x\\le |x|[\/latex]<\/li>\r\n\t
    3. 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
    4. 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\r\n[caption id=\"attachment_1644\" align=\"alignnone\" width=\"309\"]\"The 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]<\/p>\r\n<\/section>\r\n

      \r\n

      [ohm_question hide_question_number=1]204232[\/ohm_question]<\/p>\r\n<\/section>","rendered":"

      The Squeeze Theorem<\/h2>\n

      The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions.<\/p>\n

      The next theorem, called the squeeze theorem<\/strong>, proves very useful for establishing basic trigonometric limits.<\/p>\n

      This theorem allows us to calculate limits by \u201csqueezing\u201d 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].\u00a0<\/p>\n

      \"A
      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
      \n

      The Squeeze Theorem<\/h3>\n

      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

      [latex]g(x)\\le f(x)\\le h(x)[\/latex]<\/div>\n

      for all [latex]x\\ne a[\/latex] in an open interval containing [latex]a[\/latex] and<\/p>\n

      [latex]\\underset{x\\to a}{\\lim}g(x)=L=\\underset{x\\to a}{\\lim}h(x)[\/latex]<\/div>\n

      where [latex]L[\/latex] is a real number, then [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex].<\/p>\n<\/section>\n

      \n

      How To: Solve Trigonometric Limits Using the Squeeze Theorem<\/strong><\/p>\n

        \n
      1. Confirm the function shows an indeterminate form that the Squeeze Theorem can address.<\/li>\n
      2. 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
      3. Ensure [latex]g(x)[\/latex] and [latex]h(x)[\/latex] approach the same limit at the point of interest.<\/li>\n
      4. 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
        \n

        Apply the Squeeze Theorem to evaluate the limit [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}[\/latex].<\/p>\n

        First start by identify bounding functions. We know that [latex]-1\\le \\sin{x^2}\\le 1[\/latex].<\/p>\n

        Next, divide these inequalities by [latex]x[\/latex].<\/p>\n

        [latex]\\frac{-1}{x}\\le \\sin{x^2}\\le \\frac{1}{x}[\/latex]<\/p>\n

        Now, evaluate the limits of the bounding functions as [latex]x[\/latex] approaches [latex]0[\/latex].<\/p>\n

        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

        [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin{x^2}}{x}=0[\/latex]<\/p>\n

        due to the squeeze theorem.<\/p>\n<\/section>\n

        \n

        Apply the Squeeze Theorem to evaluate [latex]\\underset{x\\to 0}{\\lim}x \\cos x[\/latex].<\/p>\n