{"id":2860,"date":"2025-08-13T20:28:59","date_gmt":"2025-08-13T20:28:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2860"},"modified":"2025-08-14T16:38:54","modified_gmt":"2025-08-14T16:38:54","slug":"properties-of-limits-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/properties-of-limits-learn-it-1\/","title":{"raw":"Properties of Limits: Learn It 2","rendered":"Properties of Limits: Learn It 2"},"content":{"raw":"

Finding the Limit of a Polynomial<\/h2>\r\nNot all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit<\/strong> of a polynomial function as [latex]x[\/latex] approaches [latex]a[\/latex] is equivalent to simply evaluating the function for [latex]a[\/latex].\r\n\r\n
How To: Given a function containing a polynomial, find its limit.<\/strong>\r\n
    \r\n \t
  1. Use the properties of limits to break up the polynomial into individual terms.<\/li>\r\n \t
  2. Find the limits of the individual terms.<\/li>\r\n \t
  3. Add the limits together.<\/li>\r\n \t
  4. Alternatively, evaluate the function for [latex]a[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
    Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)[\/latex].[reveal-answer q=\"594542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594542\"]\r\n

    [latex]\\begin{align}\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)&=5\\underset{x\\to 3}{\\mathrm{lim}}\\left({x}^{2}\\right)&& \\text{Constant times a function property} \\\\ &=5\\left({3}^{2}\\right)&& \\text{Function raised to an exponent property} \\\\ &=45 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Evaluate [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left({x}^{3}-5\\right)[\/latex].[reveal-answer q=\"783831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783831\"]59[\/hidden-answer]<\/section>
    Evaluate [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)[\/latex].[reveal-answer q=\"100669\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"100669\"]\r\n

    [latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}-3x+1\\right)&=\\underset{x\\to 5}{\\mathrm{lim}}\\left(2{x}^{3}\\right)-\\underset{x\\to 5}{\\mathrm{lim}}\\left(3x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&& \\text{Sum of functions} \\\\ &=\\underset{x\\to 5}{2\\mathrm{lim}}\\left({x}^{3}\\right)-\\underset{x\\to 5}{3\\mathrm{lim}}\\left(x\\right)+\\underset{x\\to 5}{\\mathrm{lim}}\\left(1\\right)&& \\text{Constant times a function} \\\\ &=2\\left({5}^{3}\\right)-3\\left(5\\right)+1&& \\text{Function raised to an exponent} \\\\ &=236&& \\text{Evaluate} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Evaluate the following limit: [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\left({x}^{4}-4{x}^{3}+5\\right)[\/latex].[reveal-answer q=\"535807\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"535807\"]10[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]174090[\/ohm_question]<\/section>\r\n

    Finding the Limit of a Power or a Root<\/h2>\r\nWhen a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit<\/strong> of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.\r\n\r\n
    Evaluate [latex]\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}[\/latex].[reveal-answer q=\"887017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"887017\"]\r\n\r\nWe will take the limit of the function as [latex]x[\/latex] approaches 2 and raise the result to the 5th<\/sup> power.\r\n

    [latex]\\begin{align}\\underset{x\\to 2}{\\mathrm{lim}}{\\left(3x+1\\right)}^{5}&={\\left(\\underset{x\\to 2}{\\mathrm{lim}}\\left(3x+1\\right)\\right)}^{5} \\\\ &={\\left(3\\left(2\\right)+1\\right)}^{5} \\\\ &={7}^{5} \\\\ &=\\text{16,807} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Evaluate the following limit: [latex]\\underset{x\\to -4}{\\mathrm{lim}}{\\left(10x+36\\right)}^{3}[\/latex].[reveal-answer q=\"411460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411460\"]-64[\/hidden-answer]<\/section>
    Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function. For example in [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\frac{{x}^{2}+6x+8}{x - 2}\\right)[\/latex]<\/section>","rendered":"

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

    Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit<\/strong> of a polynomial function as [latex]x[\/latex] approaches [latex]a[\/latex] is equivalent to simply evaluating the function for [latex]a[\/latex].<\/p>\n

    How To: Given a function containing a polynomial, find its limit.<\/strong><\/p>\n
      \n
    1. Use the properties of limits to break up the polynomial into individual terms.<\/li>\n
    2. Find the limits of the individual terms.<\/li>\n
    3. Add the limits together.<\/li>\n
    4. Alternatively, evaluate the function for [latex]a[\/latex].<\/li>\n<\/ol>\n<\/section>\n
      Evaluate [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(5{x}^{2}\\right)[\/latex].<\/p>\n