{"id":2863,"date":"2025-08-13T20:30:12","date_gmt":"2025-08-13T20:30:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2863"},"modified":"2025-08-13T20:30:47","modified_gmt":"2025-08-13T20:30:47","slug":"properties-of-limits-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/properties-of-limits-learn-it-4\/","title":{"raw":"Properties of Limits: Learn It 4","rendered":"Properties of Limits: Learn It 4"},"content":{"raw":"
How To: Given a limit of a function containing a root, use a conjugate to evaluate.<\/strong>\r\n
    \r\n \t
  1. If the quotient as given is not in indeterminate [latex]\\left(\\frac{0}{0}\\right)[\/latex] form, evaluate directly.<\/li>\r\n \t
  2. Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD)<\/strong>.<\/li>\r\n \t
  3. If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the conjugate<\/strong> of the numerator. Recall that [latex]a\\pm \\sqrt{b}[\/latex] are conjugates.<\/li>\r\n \t
  4. Simplify.<\/li>\r\n \t
  5. Evaluate the resulting limit.<\/li>\r\n<\/ol>\r\n<\/section>
    Evaluate [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{25-x}-5}{x}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"167031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"167031\"]\r\n

    [latex]\\begin{align}\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\sqrt{25-x}-5}{x}\\right)&=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(\\sqrt{25-x}-5\\right)}{x}\\cdot \\frac{\\left(\\sqrt{25-x}+5\\right)}{\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Multiply numerator and denominator by the conjugate}. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{\\left(25-x\\right)-25}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Multiply: }\\left(\\sqrt{25-x}-5\\right)\\cdot \\left(\\sqrt{25-x}+5\\right)=\\left(25-x\\right)-25. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-x}{x\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Combine like terms}. \\\\ &=\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\frac{-\\cancel{x}}{\\cancel{x}\\left(\\sqrt{25-x}+5\\right)}\\right)&& \\text{Simplify }\\frac{-x}{x}=-1. \\\\ &=\\frac{-1}{\\sqrt{25 - 0}+5}&& \\text{Evaluate}. \\\\ &=\\frac{-1}{5+5}=-\\frac{1}{10} \\end{align}[\/latex]<\/p>\r\n\r\n

    Analysis of the Solution<\/h4>\r\nWhen determining a limit<\/strong> of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Evaluate the following limit: [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{16-h}-4}{h}\\right)[\/latex].[reveal-answer q=\"112150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"112150\"][latex]-\\frac{1}{8}[\/latex][\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]174096[\/ohm_question]<\/section>
    Evaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{4-x}{\\sqrt{x}-2}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"376512\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"376512\"]\r\n

    [latex]\\begin{align}\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{4-x}{\\sqrt{x}-2}\\right)&=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\left(2-\\sqrt{x}\\right)}{\\sqrt{x}-2}\\right)&& \\text{Factor.} \\\\ &=\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\frac{\\left(2+\\sqrt{x}\\right)\\cancel{\\left(2-\\sqrt{x}\\right)}}{-\\cancel{\\left(2-\\sqrt{x}\\right)}}\\right)&& \\text{Factor }-1\\text{ out of the denominator and implify}. \\\\ &=\\underset{x\\to 4}{\\mathrm{lim}}-\\left(2+\\sqrt{x}\\right)&& \\text{Evaluate}. \\\\ &=-\\left(2+\\sqrt{4}\\right) \\\\ &=-4 \\end{align}[\/latex]<\/p>\r\n\r\n

    Analysis of the Solution<\/h4>\r\nMultiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form\r\n

    [latex]{a}^{2}-{b}^{2}[\/latex]<\/p>\r\nand may be factored as\r\n

    [latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Evaluate the following limit: [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{x - 3}{\\sqrt{x}-\\sqrt{3}}\\right)[\/latex].[reveal-answer q=\"591987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"591987\"][latex]2\\sqrt{3}[\/latex][\/hidden-answer]<\/section>
    How To: Given a quotient with absolute values, evaluate its limit.<\/strong>\r\n
      \r\n \t
    1. Try factoring or finding the LCD.<\/li>\r\n \t
    2. If the limit<\/strong> cannot be found, choose several values close to and on either side of the input where the function is undefined.<\/li>\r\n \t
    3. Use the numeric evidence to estimate the limits on both sides.<\/li>\r\n<\/ol>\r\n<\/section>
      Evaluate [latex]\\underset{x\\to 7}{\\mathrm{lim}}\\dfrac{|x - 7|}{x - 7}[\/latex].\r\n\r\n[reveal-answer q=\"889215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"889215\"]\r\n\r\nThe function is undefined at [latex]x=7[\/latex], so we will try values close to 7 from the left and the right.\r\n\r\nLeft-hand limit: [latex]\\frac{|6.9 - 7|}{6.9 - 7}=\\frac{|6.99 - 7|}{6.99 - 7}=\\frac{|6.999 - 7|}{6.999 - 7}=-1[\/latex]\r\n\r\nRight-hand limit: [latex]\\frac{|7.1 - 7|}{7.1 - 7}=\\frac{|7.01 - 7|}{7.01 - 7}=\\frac{|7.001 - 7|}{7.001 - 7}=1[\/latex]\r\n\r\nSince the left- and right-hand limits are not equal, there is no limit.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      Evaluate [latex]\\underset{x\\to {6}^{+}}{\\mathrm{lim}}\\dfrac{6-x}{|x - 6|}[\/latex].[reveal-answer q=\"496056\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"496056\"]-1[\/hidden-answer]<\/section><\/section>","rendered":"
      How To: Given a limit of a function containing a root, use a conjugate to evaluate.<\/strong><\/p>\n
        \n
      1. If the quotient as given is not in indeterminate [latex]\\left(\\frac{0}{0}\\right)[\/latex] form, evaluate directly.<\/li>\n
      2. Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD)<\/strong>.<\/li>\n
      3. If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the conjugate<\/strong> of the numerator. Recall that [latex]a\\pm \\sqrt{b}[\/latex] are conjugates.<\/li>\n
      4. Simplify.<\/li>\n
      5. Evaluate the resulting limit.<\/li>\n<\/ol>\n<\/section>\n
        Evaluate [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{25-x}-5}{x}\\right)[\/latex].<\/p>\n