{"id":2862,"date":"2025-08-13T20:30:07","date_gmt":"2025-08-13T20:30:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2862"},"modified":"2025-08-14T16:39:01","modified_gmt":"2025-08-14T16:39:01","slug":"properties-of-limits-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/properties-of-limits-learn-it-3\/","title":{"raw":"Properties of Limits: Learn It 3","rendered":"Properties of Limits: Learn It 3"},"content":{"raw":"

Finding the Limit of a Quotient<\/h2>\r\nFinding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.\r\n\r\n
How To: Given the limit of a function in quotient form, use factoring to evaluate it.<\/strong>\r\n
    \r\n \t
  1. Factor the numerator and denominator completely.<\/li>\r\n \t
  2. Simplify by dividing any factors common to the numerator and denominator.<\/li>\r\n \t
  3. Evaluate the resulting limit, remembering to use the correct domain.<\/li>\r\n<\/ol>\r\n<\/section>
    Evaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-6x+8}{x - 2}\\right)[\/latex].[reveal-answer q=\"592817\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"592817\"]\r\n\r\nFactor where possible, and simplify.\r\n

    [latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}-6x+8}{x - 2}\\right)&=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\left(x - 2\\right)\\left(x - 4\\right)}{x - 2}\\right)&& \\text{Factor the numerator}. \\\\ &=\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{\\cancel{\\left(x - 2\\right)}\\left(x - 4\\right)}{\\cancel{x - 2}}\\right)&& \\text{Cancel the common factors}. \\\\ &=\\underset{x\\to 2}{\\mathrm{lim}}\\left(x - 4\\right)&& \\text{Evaluate}. \\\\ &=2 - 4=-2 \\end{align}[\/latex]<\/p>\r\n\r\n

    Analysis of the Solution<\/h4>\r\nWhen the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.\r\n\r\nNotice, the function\r\n

    [latex]f\\left(x\\right)=\\frac{{x}^{2}-6x+8}{x - 2}[\/latex]<\/p>\r\nis equivalent to the function\r\n

    [latex]f\\left(x\\right)=x - 4,x\\ne 2[\/latex].<\/p>\r\nNotice that the limit exists even though the function is not defined at [latex]x=2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    \r\n
    \r\n\r\nEvaluate the following limit: [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-11x+28}{7-x}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"97030\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"97030\"]\r\n\r\n-3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
    [ohm_question hide_question_numbers=1]16090[\/ohm_question]<\/section>
    Evaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{x}-\\frac{1}{5}}{x - 5}\\right)[\/latex].[reveal-answer q=\"649722\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"649722\"]\r\n\r\nFind the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator.\r\n\r\n\"Multiply\r\n

    Analysis of the Solution<\/h4>\r\nWhen determining the limit<\/strong> of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Evaluate [latex]\\underset{x\\to -5}{\\mathrm{lim}}\\left(\\dfrac{\\frac{1}{5}+\\frac{1}{x}}{10+2x}\\right)[\/latex].[reveal-answer q=\"934779\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"934779\"][latex]-\\frac{1}{50}[\/latex][\/hidden-answer]<\/section>","rendered":"

    Finding the Limit of a Quotient<\/h2>\n

    Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.<\/p>\n

    How To: Given the limit of a function in quotient form, use factoring to evaluate it.<\/strong><\/p>\n
      \n
    1. Factor the numerator and denominator completely.<\/li>\n
    2. Simplify by dividing any factors common to the numerator and denominator.<\/li>\n
    3. Evaluate the resulting limit, remembering to use the correct domain.<\/li>\n<\/ol>\n<\/section>\n
      Evaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-6x+8}{x - 2}\\right)[\/latex].<\/p>\n