🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus with Limits ## Ron Larson ## Chapter 12 ## Limits and an Introduction to Calculus ## Educators   ### Problem 1 If$f(x)$becomes arbitrarily close to a unique number$L$as$x$approaches$c$from either side, the _______ of$f(x)$as$x$approach$c$is$L$. Sheryl E. Numerade Educator ### Problem 2 An alternative notation for$\lim_{x\to c}f(x) = L$is$f(x) \rightarrow L $as$x \rightarrow c $, which is read as "$f(x)$_______$L$as$x$_______$c$". Anthony P. Numerade Educator ### Problem 3 The limit of$f(x)$as$x \rightarrow c $does not exist if$f(x)$_______ between two fixed values. Amy J. Numerade Educator ### Problem 4 To evaluate the limit of a polynomial function, use _______ _______. Anthony P. Numerade Educator ### Problem 5 GEOMETRY You create an open box from a square piece of material 24 centimeters on a side. You cut equal squares from the corners and turn up the sides. (a) Draw and label a diagram that represents the box. (b) Verify that the volume$V$of the box is given by$V=4x(12-x)^2$. (C) The box has a maximum volume when$x=4$. Use a graphing utility to complete the table and observe the behavior of the function as$x$approaches 4. Use the table to find$\lim_{x \to 4} V$. (d) Use a graphing utility to graph the volume function. Verify that the volume is maximum when$x=4$. Check back soon! ### Problem 6 GEOMETRY You are given wire and are asked to forma right triangle with a hypotenuse of$\sqrt{18}$inches whose area is as large as possible. (a) Draw and label a diagram that shows the base$x$and height$y$of the triangle. (b) Verify that the area$A$of the triangle is given by$A=\frac{1}{2}x \sqrt{18-x^{2}}$. (c) The triangle has a maximum area when$x=3$inches. Use a graphing utility to complete the table and observe the behavior of the function as$x$approaches 3. Use the table to find$\lim_{x \to 3} A$. (d) Use a graphing utility to graph the area function.Verify that the area is maximum when$x=3$inches. Anthony P. Numerade Educator ### Problem 7 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 2}\ (5x+4)$$ Amy J. Numerade Educator ### Problem 8 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 1}\ (2x^2+x-4)$$ Anthony P. Numerade Educator ### Problem 9 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 3}\ \dfrac{x-3}{x^2 -9}$$ Amy J. Numerade Educator ### Problem 10 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to -1}\ \dfrac{x+1}{x^2 -x-2}$$ Anthony P. Numerade Educator ### Problem 11 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 0}\ \dfrac{\sin\ 2x}{x}$$ Amy J. Numerade Educator ### Problem 12 In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached. $$\lim_{x \to 0}\ \dfrac{\tan\ x}{2x}$$ Anthony P. Numerade Educator ### Problem 13 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 1} \dfrac{x-1}{x^2+2x-3}$$ Amy J. Numerade Educator ### Problem 14 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -2} \dfrac{x+2}{x^2+5x+6}$$ Anthony P. Numerade Educator ### Problem 15 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{\sqrt{x+5} - \sqrt{5}}{x}$$ Amy J. Numerade Educator ### Problem 16 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -3} \dfrac{\sqrt{1-x} - 2}{x+3}$$ Anthony P. Numerade Educator ### Problem 17 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to -4} \dfrac{\dfrac{x}{x+2}-2}{x+4}$$ Amy J. Numerade Educator ### Problem 18 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 2} \dfrac{\dfrac{1}{x+2}-\dfrac{1}{4}}{x-2}$$ Anthony P. Numerade Educator ### Problem 19 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{\sin\ x}{x}$$ Amy J. Numerade Educator ### Problem 20 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{\cos\ x -1}{x}$$ Anthony P. Numerade Educator ### Problem 21 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{\sin^2\ x}{x}$$ Amy J. Numerade Educator ### Problem 22 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{2x}{\tan\ 4x}$$ Anthony P. Numerade Educator ### Problem 23 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{e^{2x} -1}{2x}$$ Amy J. Numerade Educator ### Problem 24 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 0} \dfrac{1-e^{-4x}}{x}$$ Anthony P. Numerade Educator ### Problem 25 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 1} \dfrac{\textrm{ln}(2x-1)}{x-1}$$ Amy J. Numerade Educator ### Problem 26 In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim_{x \to 1} \dfrac{\textrm{ln}(x^2)}{x-1}$$ Anthony P. Numerade Educator ### Problem 27 In Exercises 27 and 28, graph the function and find the limit(if it exists) as$x$approaches 2. $f(x)= \left\{ \begin{array}{rr} 2x + 1.& \mbox{if x < 2};\\ x + 1.& \mbox{if x \geq 2}.\end{array} \right.$ Amy J. Numerade Educator ### Problem 28 In Exercises 27 and 28, graph the function and find the limit(if it exists) as$x$approaches 2. $f(x)= \left\{ \begin{array}{rr} 2x,& \mbox{if x \leq 2};\\ x^2 -4x+1,& \mbox{if x > 2}.\end{array} \right.$ Anthony P. Numerade Educator ### Problem 29 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to -4} (x^2-3)$$ Amy J. Numerade Educator ### Problem 30 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to 2} \dfrac{3x^{2}-2}{x-2}$$ Anthony P. Numerade Educator ### Problem 31 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to -2} \dfrac{|x+2|}{x+2}$$ Amy J. Numerade Educator ### Problem 32 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to 1} \dfrac{|x-1|}{x-1}$$ Anthony P. Numerade Educator ### Problem 33 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to -2} \dfrac{x-2}{x^{2}-4}$$ Amy J. Numerade Educator ### Problem 34 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to 1} \dfrac{1}{x-1}$$ Anthony P. Numerade Educator ### Problem 35 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to 0}\ 2 \cos\dfrac{\pi}{x}$$ Amy J. Numerade Educator ### Problem 36 In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$\lim_{x \to -1}\ \sin\dfrac{\pi x}{2}$$ Anthony P. Numerade Educator ### Problem 37 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{5}{2+e^{1/x}}, \quad \lim_{x \to 0} f(x)$$ Amy J. Numerade Educator ### Problem 38 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \textrm{ln}(7-x), \quad \lim_{x \to -1} f(x)$$ Anthony P. Numerade Educator ### Problem 39 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \cos \dfrac{1}{x}, \quad \lim_{x \to 0} f(x)$$ Amy J. Numerade Educator ### Problem 40 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \sin \pi x, \quad \lim_{x \to 1} f(x)$$ Anthony P. Numerade Educator ### Problem 41 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{\sqrt{x+3}-1}{x-4}, \quad \lim_{x \to 4} f(x)$$ Amy J. Numerade Educator ### Problem 42 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{\sqrt{x+5}-4}{x-2}, \quad \lim_{x \to 2} f(x)$$ Anthony P. Numerade Educator ### Problem 43 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{x-1}{x^2 -4x+3}, \quad \lim_{x \to 1} f(x)$$ Amy J. Numerade Educator ### Problem 44 In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why. $$f(x) = \dfrac{7}{x-3}, \quad \lim_{x \to 3} f(x)$$ Anthony P. Numerade Educator ### Problem 45 In Exercises 45 and 46, use the given information to evaluate each limit. $$\lim_{x \to c} f(x)=3, \quad \lim_{x \to c} g(x)=6$$ $$\textrm{(a)} \quad \lim_{x \to c}\ [-2g(x)] \quad \quad \quad \quad \quad \textrm{(b)} \lim_{x \to c}\ [f(x)+g(x)]$$ $$\textrm{(c)} \quad \lim_{x \to c}\ \dfrac{f(x)}{g(x)} \quad \quad \quad \quad \quad \textrm{(d)} \lim_{x \to c}\ \sqrt{f(x)}$$ Amy J. Numerade Educator ### Problem 46 In Exercises 45 and 46, use the given information to evaluate each limit. $$\lim_{x \to c} f(x)=5, \quad \lim_{x \to c} g(x)=-2$$ $$\textrm{(a)} \quad \lim_{x \to c}\ [f(x)+g(x)]^2 \quad \quad \quad \quad \quad \textrm{(b)} \lim_{x \to c}\ [6f(x)g(x)]$$ $$\textrm{(c)} \quad \lim_{x \to c}\ \dfrac{5g(x)}{4f(x)} \quad \quad \quad \quad \quad \textrm{(d)} \lim_{x \to c}\ \dfrac{1}{\sqrt{f(x)}}$$ Anthony P. Numerade Educator ### Problem 47 In Exercises 47 and 48, find $$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$ $$f(x)=x^3, \quad \quad g(x)=\dfrac{\sqrt{x^2 +5}}{2x^2}$$ Amy J. Numerade Educator ### Problem 48 In Exercises 47 and 48, find $$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$ $$f(x)=\dfrac{x}{3-x}, \quad \quad g(x)=\sin \pi x$$ Anthony P. Numerade Educator ### Problem 49 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 5}\ (10-x^{2})$$ Amy J. Numerade Educator ### Problem 50 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -2}\ \left(\frac{1}{2}x^{3}-5x \right)$$ Anthony P. Numerade Educator ### Problem 51 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -3}\ (2x^2 +4x+1)$$ Amy J. Numerade Educator ### Problem 52 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -2}\ (x^3 -6x+5)$$ Anthony P. Numerade Educator ### Problem 53 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 3}\ (-\dfrac{9}{x})$$ Amy J. Numerade Educator ### Problem 54 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -5}\ \dfrac{6}{x+2}$$ Anthony P. Numerade Educator ### Problem 55 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -3}\ \dfrac{3x}{x^2 +1}$$ Amy J. Numerade Educator ### Problem 56 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 4}\ \dfrac{x-1}{x^2 +2x+3}$$ Anthony P. Numerade Educator ### Problem 57 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -2}\ \dfrac{5x+3}{2x-9}$$ Amy J. Numerade Educator ### Problem 58 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 3}\ \dfrac{x^2+1}{x}$$ Anthony P. Numerade Educator ### Problem 59 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to -1}\ \sqrt{x+2}$$ Amy J. Numerade Educator ### Problem 60 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 3}\ \sqrt{x^2-1}$$ Anthony P. Numerade Educator ### Problem 61 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 7}\ \dfrac{5x}{\sqrt{x+2}}$$ Amy J. Numerade Educator ### Problem 62 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 8}\ \dfrac{\sqrt{x+1}}{x-4}$$ Anthony P. Numerade Educator ### Problem 63 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 3}\ e^x$$ Amy J. Numerade Educator ### Problem 64 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 3}\ \textrm{ln}\ x$$ Anthony P. Numerade Educator ### Problem 65 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to \pi}\ \textrm{sin}\ 2x$$ Amy J. Numerade Educator ### Problem 66 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to \pi}\ \textrm{tan}\ x$$ Anthony P. Numerade Educator ### Problem 67 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 1/2}\ \textrm{arcsin}\ x$$ Amy J. Numerade Educator ### Problem 68 In Exercises 49-68, find the limit by direct substitution. $$\lim_{x \to 1}\ \textrm{arccos}\ \dfrac{x}{2}$$ Anthony P. Numerade Educator ### Problem 69 TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. The limit of a function as$x$approaches$c$does not exist if the function approaches$-3$from the left of$c$and$3$from the right of$c$. Amy J. Numerade Educator ### Problem 70 TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. The limit of the product of two functions is equal to the product of the limits of the two functions. Anthony P. Numerade Educator ### Problem 71 THINK ABOUT IT From Exercises 7-12, select a limit that can be reached and one that cannot be reached. (a) Use a graphing utility to graph the corresponding functions using a standard viewing window. Do the graphs reveal whether or not the limit can be reached? Explain. (b) Use a graphing utility to graph the corresponding functions using a$decimal$setting. Do the graphs reveal whether or not the limit can be reached? Explain. Check back soon! ### Problem 72 THINK ABOUT IT Use the results of Exercise 71 to draw a conclusion as to whether or not you can use the graph generated by a graphing utility to determine reliably if a limit can be reached. Anthony P. Numerade Educator ### Problem 73 THINK ABOUT IT (a) If$f(2)=4$, can you conclude anything about$\lim_{x \to 2} f(x)$? Explain your reasoning. (b) If$\lim_{x \to 2} f(x)=4$, can you conclude anything about$f(2)$? Explain your reasoning. Amy J. Numerade Educator ### Problem 74 WRITING Write a brief description of the meaning of the notation $$\lim_{x \to 5} f(x)=12$$. Anthony P. Numerade Educator ### Problem 75 THINK ABOUT IT Use a graphing utility to graph the tangent function. What are$\lim_{x \to 0} \textrm{tan}\ x$and$\lim_{x \to \pi/x} \textrm{tan}\ x$? What can you say about the existence of the$\lim_{x \to \pi/2} \textrm{tan}\ x$? Amy J. Numerade Educator ### Problem 76 CAPSTONE Use the graph of the function$f$to decide whether the value of the given quantity exists. If it does, find it. If not, explain why. (a)$f(0)$(b)$\lim_{x \to 0} f(x)$(c)$f(2)$(d)$\lim_{x \to 2} f(x)$ Anthony P. Numerade Educator ### Problem 77 WRITING Use a graphing utility to graph the function given by$f(x) = \dfrac{x^2 - 3x - 10}{x-5}$. Use the trace feature to approximate$\lim_{x \to 4} f(x)$. What do you think$\lim_{x \to 5} f(x)$equals? Is$f$defined at$x=5$? Does this affect the existence of the limit as$x$approaches$5\$? Amy J.