🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus for Business, Economics, Life Sciences, and Social Sciences 13th ## Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen ## Chapter 2 ## Limits and the Derivative ## Educators ### Problem 1 In Problems$1-8$, factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{2}-81$$ Dwijendra R. Numerade Educator ### Problem 2 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{2}-64$$ Dwijendra R. Numerade Educator ### Problem 3 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{2}-4 x-21$$ Dwijendra R. Numerade Educator ### Problem 4 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{2}+5 x-36$$ Dwijendra R. Numerade Educator ### Problem 5 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{3}-7 x^{2}+12 x$$ Dwijendra R. Numerade Educator ### Problem 6 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$x^{3}+15 x^{2}+50 x$$ Dwijendra R. Numerade Educator ### Problem 7 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$6 x^{2}-x-1$$ Dwijendra R. Numerade Educator ### Problem 8 Factor each polynomial into the product of first-degree factors with integer coefficients. (If necessary, review Section A.3). $$20 x^{2}+11 x-3$$ Dwijendra R. Numerade Educator ### Problem 9 In Problems$9-16,$use the graph of the function$f$shown to estimate the indicated limits and function values. $$f(-0.5)$$ Dwijendra R. Numerade Educator ### Problem 10 Use the graph of the function$f$shown to estimate the indicated limits and function values. $$f(-1.5)$$ Dwijendra R. Numerade Educator ### Problem 11 Use the graph of the function$f$shown to estimate the indicated limits and function values. $$f(1.75)$$ Dwijendra R. Numerade Educator ### Problem 12 Use the graph of the function$f$shown to estimate the indicated limits and function values. $$f(1.25)$$ Dwijendra R. Numerade Educator ### Problem 13 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 0^{-}} f(x)$(B)$\lim _{x \rightarrow 0^{+}} f(x)$(C)$\lim _{x \rightarrow 0} f(x)$(D)$f(0)$Dwijendra R. Numerade Educator ### Problem 14 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 1^{-}} f(x)$(B)$\lim _{x \rightarrow 1^{+}} f(x)$(C)$\lim _{i} f(x)$(D)$f(1)$Dwijendra R. Numerade Educator ### Problem 15 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 2^{-}} f(x)$(B)$\lim _{x \rightarrow 2^{+}} f(x)$(C)$\lim _{x \rightarrow 2} f(x)$(D)$f(2)$(E) Is it possible to redefine$f(2)$so that$\lim _{x \rightarrow 2} f(x)=f(2)$? Explain. Dwijendra R. Numerade Educator ### Problem 16 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 4^{-}} f(x)$(B)$\lim _{x \rightarrow 4^{+}} f(x)$(C)$\lim _{x \rightarrow 4} f(x)$(D)$f(4)$(E) Is it possible to define$f(4)$so that$\lim _{x \rightarrow 4} f(x)=f(4)$? Explain. Dwijendra R. Numerade Educator ### Problem 17 In Problems$17-24$, use the graph of the function$g$shown to estimate the indicated limits and function values. $$g(1.9)$$ Dwijendra R. Numerade Educator ### Problem 18 Use the graph of the function$g$shown to estimate the indicated limits and function values. $$g(0.1)$$ Dwijendra R. Numerade Educator ### Problem 19 Use the graph of the function$g$shown to estimate the indicated limits and function values. $$g(3.5)$$ Dwijendra R. Numerade Educator ### Problem 20 Use the graph of the function$g$shown to estimate the indicated limits and function values. $$g(2.5)$$ Dwijendra R. Numerade Educator ### Problem 21 Use the graph of the function$g$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 1^{-}} g(x)$(B)$\lim _{x \rightarrow 1^{+}} g(x)$(C)$\lim _{x \rightarrow 1} g(x)$(D)$g(1)$(E) Is it possible to define$g(1)$so that$\lim _{x \rightarrow 1} g(x)=g(1) ?$Explain. Dwijendra R. Numerade Educator ### Problem 22 Use the graph of the function$g$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 2^{-}} g(x)$(B)$\lim _{x \rightarrow 2^{+}} g(x)$(C)$\lim _{x \rightarrow 2} g(x)$(D)$g(2)$Dwijendra R. Numerade Educator ### Problem 23 Use the graph of the function$g$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 3^{-}} g(x)$(B)$\lim _{x \rightarrow 3^{+}} g(x)$(C)$\lim _{x \rightarrow 3} g(x)$(D)$g(3)$(E) Is it possible to redefine$g(3)$so that$\lim _{x \rightarrow 3} g(x)=g(3) ?$Explain. Dwijendra R. Numerade Educator ### Problem 24 Use the graph of the function$g$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 4^{-}} g(x)$(B)$\lim _{x \rightarrow 4^{+}} g(x)$(C)$\lim _{x \rightarrow 4} g(x)$(D)$g(4)$Dwijendra R. Numerade Educator ### Problem 25 In Problems$25-28,$use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow-3^{+}} f(x)$(B)$\lim _{x \rightarrow-3^{-}} f(x)$(C)$\lim _{x \rightarrow-3} f(x)$(D)$f(-3)$(E) Is it possible to redefine$f(-3)$so that$\lim _{x \rightarrow-3} f(x)=f(-3) ?$Explain. Dwijendra R. Numerade Educator ### Problem 26 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow-2^{+}} f(x)$(B)$\lim _{x \rightarrow-2^{-}} f(x)$(C)$\lim _{x \rightarrow-2} f(x)$(D)$f(-2)$(E) Is it possible to redefine$f(-2)$so that$\lim _{x \rightarrow-2} f(x)=f(-2) ?$Explain. Dwijendra R. Numerade Educator ### Problem 27 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 0^{+}} f(x)$(B)$\lim _{x \rightarrow 0^{-}} f(x)$(C)$\lim _{x \rightarrow 0} f(x)$(D)$f(0)$(E) Is it possible to define$f(0)$so that$\lim _{x \rightarrow 0} f(x)=f(0) ?$Explain. Dwijendra R. Numerade Educator ### Problem 28 Use the graph of the function$f$shown to estimate the indicated limits and function values. (A)$\lim _{x \rightarrow 2^{+}} f(x)$(B)$\lim _{x \rightarrow 2^{-}} f(x)$(C)$\lim _{x \rightarrow 2} f(x)$(D)$f(2)$(E) Is it possible to redefine$f(2)$so that$\lim _{x \rightarrow 2} f(x)=f(2) ?$Explain. Dwijendra R. Numerade Educator ### Problem 29 In Problems 29-38, find each limit if it exists. $$\lim _{x \rightarrow 3} 4 x$$ Dwijendra R. Numerade Educator ### Problem 30 Find each limit if it exists. $$\lim _{x \rightarrow-2} 3 x$$ Dwijendra R. Numerade Educator ### Problem 31 Find each limit if it exists. $$\lim _{x \rightarrow-4}(x+5)$$ Dwijendra R. Numerade Educator ### Problem 32 Find each limit if it exists. $$\lim _{x \rightarrow 5}(x-3)$$ Dwijendra R. Numerade Educator ### Problem 33 Find each limit if it exists. $$\lim _{x \rightarrow 2} x(x-4)$$ Dwijendra R. Numerade Educator ### Problem 34 Find each limit if it exists. $$\lim _{x \rightarrow-1} x(x+3)$$ Dwijendra R. Numerade Educator ### Problem 35 Find each limit if it exists. $$\lim _{x \rightarrow-3} \frac{x}{x+5}$$ Dwijendra R. Numerade Educator ### Problem 36 Find each limit if it exists. $$\lim _{x \rightarrow 4} \frac{x-2}{x}$$ Dwijendra R. Numerade Educator ### Problem 37 Find each limit if it exists. $$\lim _{x \rightarrow 1} \sqrt{5 x+4}$$ Dwijendra R. Numerade Educator ### Problem 38 Find each limit if it exists. $$\lim _{x \rightarrow 0} \sqrt{16-7 x}$$ Dwijendra R. Numerade Educator ### Problem 39 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits in Problems$39-46 .$$$\lim _{x \rightarrow 1}(-3) f(x)$$ Dwijendra R. Numerade Educator ### Problem 40 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1} 2 g(x)$$ Dwijendra R. Numerade Educator ### Problem 41 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1}[2 f(x)+g(x)]$$ Dwijendra R. Numerade Educator ### Problem 42 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1}[g(x)-3 f(x)]$$ Dwijendra R. Numerade Educator ### Problem 43 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1} \frac{2-f(x)}{x+g(x)}$$ Dwijendra R. Numerade Educator ### Problem 44 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1} \frac{3-f(x)}{1-4 g(x)}$$ Dwijendra R. Numerade Educator ### Problem 45 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1} \sqrt{g(x)-f(x)}$$ Dwijendra R. Numerade Educator ### Problem 46 Given that$\lim _{x \rightarrow 1} f(x)=-5$and$\lim _{x \rightarrow 1} g(x)=4,$find the indicated limits. $$\lim _{x \rightarrow 1} \sqrt[3]{2 x+2 f(x)}$$ Dwijendra R. Numerade Educator ### Problem 47 In Problems$47-50$, sketch a possible graph of a function that satisfies the given conditions. $$f(0)=1 ; \lim _{x \rightarrow 0^{-}} f(x)=3 ; \lim _{x \rightarrow 0^{+}} f(x)=1$$ Dwijendra R. Numerade Educator ### Problem 48 Sketch a possible graph of a function that satisfies the given conditions. $$f(1)=-2 ; \lim _{x \rightarrow 1^{-}} f(x)=2 ; \lim _{x \rightarrow 1^{+}} f(x)=-2$$ Check back soon! ### Problem 49 Sketch a possible graph of a function that satisfies the given conditions. $$f(-2)=2 ; \lim _{x \rightarrow-2^{-}} f(x)=1 ; \lim _{x \rightarrow-2^{+}} f(x)=1$$ Dwijendra R. Numerade Educator ### Problem 50 Sketch a possible graph of a function that satisfies the given conditions. $$f(0)=-1 ; \lim _{x \rightarrow 0^{-}} f(x)=2 ; \lim _{x \rightarrow 0^{+}} f(x)=2$$ Check back soon! ### Problem 51 In Problems$51-66$, find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{ll}1-x^{2} & \text { if } x \leq 0 \\ 1+x^{2} & \text { if } x>0\end{array}\right.$. Find (A)$\lim _{x \rightarrow 0^{+}} f(x)$(B)$\lim _{x \rightarrow 0^{-}} f(x)$(C)$\lim _{x \rightarrow 0} f(x)$(D)$f(0)$Dwijendra R. Numerade Educator ### Problem 52 Find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{ll}2+x & \text { if } x \leq 0 \\ 2-x & \text { if } x>0\end{array}\right.$. Find (A)$\lim _{x \rightarrow 0^{+}} f(x)$(B)$\lim _{x \rightarrow 0^{-}} f(x)$(C)$\lim _{x \rightarrow 0} f(x)$(D)$f(0)$Dwijendra R. Numerade Educator ### Problem 53 Find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<1 \\ 2 x & \text { if } x>1\end{array} .\right.$Find (A)$\lim _{x \rightarrow 1^{+}} f(x)$(B)$\lim _{x \rightarrow 1^{-}} f(x)$(C)$\lim _{x \rightarrow 1} f(x)$(D)$f(1)$Dwijendra R. Numerade Educator ### Problem 54 Find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{cl}x+3 & \text { if } x<-2 \\ \sqrt{x+2} & \text { if } x>-2\end{array}\right.$. Find (A)$\lim _{x \rightarrow-2^{+}} f(x)$(B)$\lim _{x \rightarrow-2^{-}} f(x)$(C)$\lim _{x \rightarrow-2} f(x)$(D)$f(-2)$Dwijendra R. Numerade Educator ### Problem 55 Find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x+3} & \text { if } x<0 \\ \frac{x^{2}-9}{x-3} & \text { if } x>0\end{array}\right.$. Find (A)$\lim _{x \rightarrow-3} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$Dwijendra R. Numerade Educator ### Problem 56 Find each indicated quantity if it exists. Let$f(x)=\left\{\begin{array}{ll}\frac{x}{x+3} & \text { if } x<0 \\ \frac{x}{x-3} & \text { if } x>0\end{array}\right.$. Find (A)$\lim _{x \rightarrow-3} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$Dwijendra R. Numerade Educator ### Problem 57 Find each indicated quantity if it exists. Let$f(x)=\frac{|x-1|}{x-1} .$Find (A)$\lim _{x \rightarrow 1^{+}} f(x)$(B)$\lim _{x \rightarrow 1^{-}} f(x)$(C)$\lim _{x \rightarrow 1} f(x)$(D)$f(1)$Dwijendra R. Numerade Educator ### Problem 58 Find each indicated quantity if it exists. Let$f(x)=\frac{x-3}{|x-3|} .$Find (A)$\lim _{x \rightarrow 3^{+}} f(x)$(B)$\lim _{x \rightarrow 3^{-}} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$(D)$f(3)$Dwijendra R. Numerade Educator ### Problem 59 Let$f(x)=\frac{x-3}{|x-3|} .$Find (A)$\lim _{x \rightarrow 3^{+}} f(x)$(B)$\lim _{x \rightarrow 3^{-}} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$(D)$f(3)$Dwijendra R. Numerade Educator ### Problem 60 Find each indicated quantity if it exists. Let$f(x)=\frac{x+3}{x^{2}+3 x} .$Find (A)$\lim _{x \rightarrow-3} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$Dwijendra R. Numerade Educator ### Problem 61 Find each indicated quantity if it exists. Let$f(x)=\frac{x^{2}-x-6}{x+2} .$Find (A)$\lim _{x \rightarrow-2} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 3} f(x)$Dwijendra R. Numerade Educator ### Problem 62 Find each indicated quantity if it exists. Let$f(x)=\frac{x^{2}+x-6}{x+3}$. Find (A)$\lim _{x \rightarrow-3} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 2} f(x)$Dwijendra R. Numerade Educator ### Problem 63 Find each indicated quantity if it exists. Let$f(x)=\frac{(x+2)^{2}}{x^{2}-4}$. Find (A)$\lim _{x \rightarrow-2} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 2} f(x)$Dwijendra R. Numerade Educator ### Problem 64 Find each indicated quantity if it exists. Let$f(x)=\frac{x^{2}-1}{(x+1)^{2}} .$Find (A)$\lim _{x \rightarrow-1} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 1} f(x)$Dwijendra R. Numerade Educator ### Problem 65 Find each indicated quantity if it exists. Let$f(x)=\frac{2 x^{2}-3 x-2}{x^{2}+x-6} .$Find (A)$\lim _{x \rightarrow 2} f(x)$(B)$\lim _{x \rightarrow 0} f(x)$(C)$\lim _{x \rightarrow 1} f(x)$Dwijendra R. Numerade Educator ### Problem 66 Find each indicated quantity if it exists. Let$f(x)=\frac{3 x^{2}+2 x-1}{x^{2}+3 x+2}$. Find (A)$\lim _{x \rightarrow-3} f(x)$(B)$\lim _{x \rightarrow-1} f(x)$(C)$\lim _{x \rightarrow 2} f(x)$Dwijendra R. Numerade Educator ### Problem 67 In Problems 67-72, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$\lim _{x \rightarrow 1} f(x)=0$and$\lim _{x \rightarrow 1} g(x)=0,$then$\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}=0$. Dwijendra R. Numerade Educator ### Problem 68 Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$\lim _{x \rightarrow 1} f(x)=1$and$\lim _{x \rightarrow 1} g(x)=1,$then$\lim _{x \rightarrow 1} \frac{f(x)}{g(x)}=1$. Dwijendra R. Numerade Educator ### Problem 69 Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$f$is a polynomial, then, as$x$approaches 0 , the right-hand limit exists and is equal to the left-hand limit. Dwijendra R. Numerade Educator ### Problem 70 Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$f$is a rational function, then, as$x$approaches 0 , the right hand limit exists and is equal to the left-hand limit. Check back soon! ### Problem 71 Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$f$is a function such that$\lim _{x \rightarrow 0} f(x)$exists, then$f(0)$exists. Check back soon! ### Problem 72 Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If$f$is a function such that$f(0)$exists, then$\lim _{x \rightarrow 0} f(x)$exists. Check back soon! ### Problem 73 In Problems$73-80$, is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 7} \frac{(x-7)^{2}}{x^{2}-4 x-21}$$ Dwijendra R. Numerade Educator ### Problem 74 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 2} \frac{x-5}{x+2}$$ Dwijendra R. Numerade Educator ### Problem 75 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 4} \frac{x^{2}+4}{(x+4)^{2}}$$ Dwijendra R. Numerade Educator ### Problem 76 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 9} \frac{x^{2}-5 x-36}{x-9}$$ Dwijendra R. Numerade Educator ### Problem 77 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow-6} \frac{x^{2}+36}{x+6}$$ Dwijendra R. Numerade Educator ### Problem 78 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 10} \frac{x^{2}-15 x+50}{(x-10)^{2}}$$ Dwijendra R. Numerade Educator ### Problem 79 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow 8} \frac{x-8}{x^{2}-64}$$ Dwijendra R. Numerade Educator ### Problem 80 Is the limit expression a$0 / 0$indeterminate form? Find the limit or explain why the limit does not exist. $$\lim _{x \rightarrow-3} \frac{x+3}{x-3}$$ Dwijendra R. Numerade Educator ### Problem 81 Compute the following limit for each function in Problems$81-88 .$$$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=3 x+1$$ Dwijendra R. Numerade Educator ### Problem 82 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=5 x-1$$ Dwijendra R. Numerade Educator ### Problem 83 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{2}+1$$ Dwijendra R. Numerade Educator ### Problem 84 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=x^{2}-2$$ Dwijendra R. Numerade Educator ### Problem 85 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=-7 x+9$$ Dwijendra R. Numerade Educator ### Problem 86 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=-4 x+13$$ Dwijendra R. Numerade Educator ### Problem 87 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=|x+1|$$ Dwijendra R. Numerade Educator ### Problem 88 Compute the following limit for each function. $$\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$$ $$f(x)=-3|x|$$ Dwijendra R. Numerade Educator ### Problem 89 Let$f$be defined by $$f(x)=\left\{\begin{array}{ll}1+m x & \text { if } x \leq 1 \\4-m x & \text { if } x>1\end{array}\right.$$ where$m$is a constant. (A) Graph$f$for$m=1,$and find $$\lim _{x \rightarrow 1^{-}} f(x) \quad \text { and } \quad \lim _{x \rightarrow 1^{+}} f(x)$$ (B) Graph$f$for$m=2,$and find $$\lim _{x \rightarrow 1^{-}} f(x) \quad \text { and } \quad \lim _{x \rightarrow 1^{+}} f(x)$$ (C) Find$m$so that $$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$$ and graph$f$for this value of$m$. (D) Write a brief verbal description of each graph. How does the graph in part (C) differ from the graphs in parts (A) and$(\mathrm{B}) ?$Check back soon! ### Problem 90 Let$f$be defined by $$f(x)=\left\{\begin{array}{cl}-3 m+0.5 x & \text { if } x \leq 2 \\3 m-x & \text { if } x>2\end{array}\right.$$ where$m$is a constant. (A) Graph$f$for$m=0$, and find $$\lim _{x \rightarrow 2^{-}} f(x) \text { and } \lim _{x \rightarrow 2^{+}} f(x)$$ (B) Graph$f$for$m=1$, and find $$\lim _{x \rightarrow 2^{-}} f(x) \text { and } \lim _{x \rightarrow+} f(x)$$ (C) Find$m$so that $$\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)$$ and graph$f$for this value of$m$. (D) Write a brief verbal description of each graph. How does the graph in part (C) differ from the graphs in parts (A) and (B)? Check back soon! ### Problem 91 A long-distance telephone service charges$\$0.99$ for the first 20 minutes or less of a call and $\$ 0.07$per minute thereafter. (A) Write a piecewise definition of the charge$F(x)$for a long-distance call lasting$x$minutes. (B) Graph$F(x)$for$0<x \leq 40$. (C) Find$\lim _{x \rightarrow 20^{-}} F(x), \lim _{x \rightarrow 20^{+}} F(x),$and$\lim _{x \rightarrow 20} F(x),$which- ever exist. Check back soon! ### Problem 92 A second long-distance telephone service charges$\$0.09$ per minute for calls lasting 10 minutes or more and $\$ 0.18$per minute for calls lasting less than 10 minutes. (A) Write a piecewise definition of the charge$G(x)$for a long-distance call lasting$x$minutes. (B) Graph$G(x)$for$0<x \leq 40$(C) Find$\lim _{x \rightarrow 10^{-}} G(x), \lim _{x \rightarrow 10^{+}} G(x),$and$\lim _{x \rightarrow 10} G(x),$which-ever exist. Check back soon! ### Problem 93 Refer to Problems 91 and$92 .$Write a brief verbal comparison of the two services described for calls lasting 20 minutes or less. Check back soon! ### Problem 94 Refer to Problems 91 and$92 .$Write a brief verbal comparison of the two services described for calls lasting more than 20 minutes. Check back soon! ### Problem 95 A company sells custom embroidered apparel and promotional products. Table 1 shows the volume discounts offered by the company, where$x$is the volume of a purchase in dollars. Problems 95 and 96 deal with two different interpretations of this discount method. Assume that the volume discounts in Table 1 apply to the entire purchase. That is, if the volume$x$satisfies$\$300 \leq x<\$ 1,000,$then the entire purchase is discounted$3 \%$. If the volume$x$satisfies$\$1,000 \leq x<\$ 3,000$the entire purchase is discounted$5 \%,$and so on. (A) If$x$is the volume of a purchase before the discount is applied, then write a piecewise definition for the discounted price$D(x)$of this purchase. (B) Use one-sided limits to investigate the limit of$D(x)$as$x$approaches$\$1,000 .$ As $x$ approaches $\$ 3,000$Check back soon! ### Problem 96 A company sells custom embroidered apparel and promotional products. Table 1 shows the volume discounts offered by the company, where$x$is the volume of a purchase in dollars. deal with two different interpretations of this discount method. Assume that the volume discounts in Table 1 apply only to that portion of the volume in each interval. That is, the discounted price for a$\$4,000$ purchase would be computed as follows:
$$300+0.97(700)+0.95(2,000)+0.93(1,000)=3,809$$
(A) If $x$ is the volume of a purchase before the discount is applied, then write a piecewise definition for the discounted price $P(x)$ of this purchase.
(B) Use one-sided limits to investigate the limit of $P(x)$ as $x$ approaches $\$ 1,000 .$As$x$approaches$\$3,000$
(C) Compare this discount method with the one in Problem
95. Does one always produce a lower price than the other? Discuss.

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A state charges polluters an annual fee of $\$ 20$per ton for each ton of pollutant emitted into the atmosphere, up to a maximum of 4,000 tons. No fees are charged for emissions beyond the 4,000 -ton limit. Write a piecewise definition of the fees$F(x)$charged for the emission of$x$tons of pollutant in a year. What is the limit of$F(x)$as$x$approaches 4,000 tons? As$x$approaches 8,000 tons? Check back soon! ### Problem 98 Refer to Problem 97 . The average fee per ton of pollution is given by$A(x)=F(x) / x$. Write a piecewise definition of$A(x)$. What is the limit of$A(x)$as$x$approaches 4,000 tons? As$x$approaches 8,000 tons? Check back soon! ### Problem 99 Statisticians often use piecewise-defined functions to predict outcomes of elections. For the following functions$f$and$g,$find the limit of each function as$x$approaches 5 and as$x\$ approaches 10
$$\begin{array}{l} f(x)=\left\{\begin{array}{cl} 0 & \text { if } x \leq 5 \\ 0.8-0.08 x & \text { if } 5 < x < 10 \\ 0 & \text { if } 10 \leq x \end{array}\right. \\ g(x)=\left\{\begin{array}{cl} 0 & \text { if } x \leq 5 \\ 0.8 x-0.04 x^{2}-3 & \text { if } 5 < x < 10 \\ 1 & \text { if } 10 \leq x \end{array}\right. \end{array}$$

Dwijendra R.