Calculus Chapter 2 Limits and Continuity Answers
Problem 1
Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
\begin{equation}\begin{array}{ll}{\text { a. } \lim _{x \rightarrow-1^{+}} f(x)=1} & {\text { b. } \lim _{x \rightarrow 0^{-}} f(x)=0} \\ {\text { c. } \lim _{x \rightarrow 0^{-}} f(x)=1} & {\text { d. } \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)} \\ {\text { e. } \lim _{x \rightarrow 0} f(x) \text { exists. }} & {\text { f. } \lim _{x \rightarrow 0} f(x)=0} \\ {\text { g. } \lim _{x \rightarrow 0} f(x)=1} & {\text { h. } \lim _{x \rightarrow 1} f(x)=1} \\ {\text { i. } \lim _{x \rightarrow 1} f(x)=0} & {\text { j. } \lim _{x \rightarrow 2} f(x)=2} \\ {\text { k. } \lim _{x \rightarrow-1} f(x) \text { does not exist. }} & {\text { l. } \lim _{x \rightarrow 2^{+}} f(x)=0} \end{array}\end{equation}
Problem 2
Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
\begin{equation}
\begin{array}{ll}{\text { a. } \lim _{x \rightarrow-1^{+}} f(x)=1} \quad \quad \quad \quad {\text { b. } \lim _{x \rightarrow 2} f(x) \text { does not exist. }} \\ {\text { c. } \lim _{x \rightarrow 2} f(x)=2} \quad \quad \quad \quad \quad {\text { d. } \lim _{x \rightarrow 1^{-}} f(x)=2} \\ {\text { e.} \lim _{x \rightarrow 1^{+}} f(x)=1} \quad \quad \quad \quad \quad {\text { f. } \lim _{x \rightarrow 1^{-}} f(x) \text { does not exist.}} \\ {\text { g. } \lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}} f(x)} \\ {\text { h. } \lim _{x \rightarrow c} f(x) \text { exists at every } c \text { in the open interval }(-1,1) \text { . }} \\ {\text { i. } \lim _{x \rightarrow c} f(x) \text { exists at every } c \text { in the open interval }(1,3) \text { . }} \\ {\text { j. } \lim _{x \rightarrow-1^{-}} f(x)=0 \quad \quad \quad \quad \text { k. } \lim _{x \rightarrow 3^{+}} f(x) \text { does not exist. }} \end{array}
\end{equation}
Matt J.
Numerade Educator
Problem 3
\begin{equation}f(x)=\left\{\begin{array}{ll}{3-x,} & {x<2} \\ {\frac{x}{2}+1,} & {x>2}.\end{array}\right.\end{equation}
\begin{equation}
\begin{array}{l}{\text { a. Find } \lim _{x \rightarrow 2^{+}} f(x) \text { and } \lim _{x \rightarrow 2^{-}} f(x).} \\ {\text { b. Does } \lim _{x \rightarrow 2^{+}} f(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { c. Find } \lim _{x \rightarrow 4^{-}} f(x) \text { and } \lim _{x \rightarrow 4^{+}} f(x).} \\ {\text { d. Does } \lim _{x \rightarrow 4} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}
\end{equation}
Problem 4
\begin{equation}
f(x)=\left\{\begin{array}{ll}{3-x,} & {x<2} \\ {2,} & {x=2} \\ {\frac{x}{2},} & {x>2}.\end{array}\right.
\end{equation}
\begin{equation}
\begin{array}{l}{\text { a. Find } \lim _{x \rightarrow 2^{+}} f(x), \lim _{x \rightarrow 2^{-}} f(x), \text { and } f(2) .} \\ {\text { b. Does } \lim _{x \rightarrow 2^{+}} f(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { c. Find } \lim _{x \rightarrow-1^{-}} f(x) \text { and } \lim _{x \rightarrow-1^{+}} f(x).} \\ {\text { d. Does } \lim _{x \rightarrow-1} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}
\end{equation}
Matt J.
Numerade Educator
Problem 5
\begin{equation}f(x)=\left\{\begin{array}{ll}{0,} & {x \leq 0} \\ {\sin \frac{1}{x},} & {x>0}.\end{array}\right.\end{equation}
\begin{equation}\begin{array}{l}{\text { a. Does } \lim _{x \rightarrow 0^{+}} f(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { b. Does } \lim _{x \rightarrow 0^{+}} f(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { c. Does } \lim _{x \rightarrow 0} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}\end{equation}
Problem 6
Let $g(x)=\sqrt{x} \sin (1 / x).$
\begin{equation}\begin{array}{l}{\text { a. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { b. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { c. Does } \lim _{x \rightarrow 0} g(x) \text { exist? If so, what is it? If not, why not? }}\end{array}\end{equation}
Matt J.
Numerade Educator
Problem 7
\begin{equation}\begin{array}{ll}{\text { a. }} {\text { Graph } f(x)=\left\{\begin{array}{ll}{x^{3},} & {x \neq 1} \\ {0,} & {x=1}.\end{array}\right.} \\ {\text { b. Find } \lim _{x \rightarrow 1^{-}} f(x) \text { and } \lim _{x \rightarrow 1^{+}} f(x)} \\ {\text { c. Does } \lim _{x \rightarrow 1} f(x) \text { exist? If so, what is it? If not, why not? }} \end{array}\end{equation}
Problem 8
\begin{equation}
\begin{array}{ll}{\text { a. }} {\text { Graph } f(x)=\left\{\begin{array}{ll}{1-x^{2},} & {x \neq 1} \\ {2,} & {x=1}.\end{array}\right.} \\ {\text { b. Find } \lim _{x \rightarrow 1^{+}} f(x) \text { and } \lim _{x \rightarrow 1}-f(x).} \\ {\text { c. Does } \lim _{x \rightarrow 1} f(x) \text { exist? If so, what is it? If not, why not? }} \end{array}
\end{equation}
Matt J.
Numerade Educator
Problem 9
Graph the functions. Then answer these questions.
\begin{equation}
\begin{array}{l}{\text { a. What are the domain and range of } f ?} \\ {\text { b. At what points } c, \text { if any, does } \lim _{x \rightarrow c} f(x) \text { exist? }} \\ {\text { c. At what points does the left-hand limit exist but not the right-hand }} \\ \quad {\text { limit? }} \\ {\text { d. At what points does the right-hand limit exist but not the left- hand}} \\ \quad {\text { limit? }}\end{array}
\end{equation}
\begin{equation}
f(x)=\left\{\begin{array}{ll}{\sqrt{1-x^{2}},} & {0 \leq x<1} \\ {1,} & {1 \leq x<2} \\ {2,} & {x=2}\end{array}\right.
\end{equation}
Leon D.
Numerade Educator
Problem 10
Graph the functions. Then answer these questions.
\begin{equation}
\begin{array}{l}{\text { a. What are the domain and range of } f ?} \\ {\text { b. At what points } c, \text { if any, does } \lim _{x \rightarrow c} f(x) \text { exist? }} \\ {\text { c. At what points does the left-hand limit exist but not the right-hand }} \\ \quad {\text { limit? }} \\ {\text { d. At what points does the right-hand limit exist but not the left- hand}} \\ \quad {\text { limit? }}\end{array}
\end{equation}
\begin{equation}
f(x)=\left\{\begin{array}{ll}{x,} & {-1 \leq x < 0, \text { or } 0 < x \leq 1} \\ {1,} & {x=0} \\ {0,} & {x<-1 \text { or } x>1}\end{array}\right.
\end{equation}
Matt J.
Numerade Educator
Problem 11
Find the limits.
\begin{equation}\lim _{x \rightarrow-0.5^{-}} \sqrt{\frac{x+2}{x+1}}\end{equation}
Problem 12
Find the limits.
\begin{equation}\lim _{x \rightarrow 1^{+}} \sqrt{\frac{x-1}{x+2}}\end{equation}
Matt J.
Numerade Educator
Problem 13
Find the limits.
\begin{equation}\lim _{x \rightarrow-2^{+}}\left(\frac{x}{x+1}\right)\left(\frac{2 x+5}{x^{2}+x}\right)\end{equation}
Problem 14
Find the limits.
\begin{equation}\lim _{x \rightarrow 1^{-}}\left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)\end{equation}
Matt J.
Numerade Educator
Problem 15
Find the limits.
\begin{equation}\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+4 h+5}-\sqrt{5}}{h}\end{equation}
Problem 16
Find the limits.
\begin{equation}\lim _{h \rightarrow 0} \frac{\sqrt{6}-\sqrt{5 h^{2}+11 h+6}}{h}\end{equation}
Matt J.
Numerade Educator
Problem 17
Find the limits.
\begin{equation}\quad \text { a. }\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2} \quad \text { b. } \lim _{x \rightarrow-2^{-}}(x+3) \frac{|x+2|}{x+2}\end{equation}
Carson M.
Numerade Educator
Problem 18
Find the limits.
\begin{equation}\quad \text { a. }\lim _{x \rightarrow 1^{+}} \frac{\sqrt{2 x}(x-1)}{|x-1|} \quad \text { b. } \lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{|x-1|}\end{equation}
Matt J.
Numerade Educator
Problem 19
Find the limits.
\begin{equation}\quad \text { a. }\lim _{x \rightarrow 0^{+}} \frac{|\sin x|}{\sin x} \quad \text { b. } \lim _{x \rightarrow 0^{-}} \frac{|\sin x|}{\sin x}\end{equation}
Problem 20
Find the limits.
\begin{equation}\quad \text { a. } \lim _{x \rightarrow 0^{+}} \frac{1-\cos x}{|\cos x-1|} \quad \text { b. } \lim _{x \rightarrow 0^{-}} \frac{\cos x-1}{|\cos x-1|}\end{equation}
Matt J.
Numerade Educator
Problem 21
Use the graph of the greatest integer function $y=\lfloor x\rfloor,$ Figure 1.10 in Section $1.1,$ to help you find the limits.
\begin{equation}\quad \text { a. }\lim _{\theta \rightarrow 3^{+}} \frac{\lfloor\theta\rfloor}{\theta} \quad \text { b. } \lim _{\theta \rightarrow 3 ^{-}} \frac{\lfloor\theta\rfloor}{\theta}\end{equation}
Problem 22
Use the graph of the greatest integer function $y=\lfloor x\rfloor,$ Figure 1.10 in Section $1.1,$ to help you find the limits.
\begin{equation}\lim _{t \rightarrow 4^{4}}(t-\lfloor t\rfloor) \quad \text { b. } \lim _{t \rightarrow 4^{-}}(t-\lfloor t\rfloor)\end{equation}
Matt J.
Numerade Educator
Problem 23
Find the limits.
\begin{equation}
\lim _{\theta \rightarrow 0} \frac{\sin \sqrt{2} \theta}{\sqrt{2} \theta}
\end{equation}
Problem 24
Find the limits.
\begin{equation}\lim _{t \rightarrow 0} \frac{\sin k t}{t} \quad(k \text { constant })\end{equation}
Matt J.
Numerade Educator
Problem 25
Find the limits.
\begin{equation}\lim _{y \rightarrow 0} \frac{\sin 3 y}{4 y}\end{equation}
Problem 26
Find the limits.
\begin{equation}\lim _{h \rightarrow 0^{-}} \frac{h}{\sin 3 h}\end{equation}
Matt J.
Numerade Educator
Problem 27
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{\tan 2 x}{x}\end{equation}
Problem 28
Find the limits.
\begin{equation}\lim _{t \rightarrow 0} \frac{2 t}{\tan t}\end{equation}
Matt J.
Numerade Educator
Problem 29
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{x \csc 2 x}{\cos 5 x}\end{equation}
Problem 30
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} 6 x^{2}(\cot x)(\csc 2 x)\end{equation}
Matt J.
Numerade Educator
Problem 31
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{x+x \cos x}{\sin x \cos x}\end{equation}
Problem 32
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{x^{2}-x+\sin x}{2 x}\end{equation}
Matt J.
Numerade Educator
Problem 33
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin 2 \theta}\end{equation}
Problem 34
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{x-x \cos x}{\sin ^{2} 3 x}\end{equation}
Matt J.
Numerade Educator
Problem 35
Find the limits.
\begin{equation}\lim _{t \rightarrow 0} \frac{\sin (1-\cos t)}{1-\cos t}\end{equation}
Problem 36
Find the limits.
\begin{equation}\lim _{h \rightarrow 0} \frac{\sin (\sin h)}{\sin h}\end{equation}
Matt J.
Numerade Educator
Problem 37
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\sin 2 \theta}\end{equation}
Problem 38
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 4 x}\end{equation}
Matt J.
Numerade Educator
Problem 39
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \theta \cos \theta\end{equation}
Problem 40
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \sin \theta \cot 2 \theta\end{equation}
Matt J.
Numerade Educator
Problem 41
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 8 x}\end{equation}
Problem 42
Find the limits.
\begin{equation}\lim _{y \rightarrow 0} \frac{\sin 3 y \cot 5 y}{y \cot 4 y}\end{equation}
Matt J.
Numerade Educator
Problem 43
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta^{2} \cot 3 \theta}\end{equation}
Problem 44
Find the limits.
\begin{equation}\lim _{\theta \rightarrow 0} \frac{\theta \cot 4 \theta}{\sin ^{2} \theta \cot ^{2} 2 \theta}\end{equation}
Matt J.
Numerade Educator
Problem 45
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{2 x}\end{equation}
Problem 46
Find the limits.
\begin{equation}\lim _{x \rightarrow 0} \frac{\cos ^{2} x-\cos x}{x^{2}}\end{equation}
Matt J.
Numerade Educator
Problem 47
Once you know $\lim _{x \rightarrow a^{+}} f(x)$ and $\lim _{x \rightarrow a^{-}} f(x)$ at an interior point of the domain of $f,$ do you then know lim $_{x \rightarrow a} f(x) ?$ Give reasons for your answer.
Problem 48
If you know that $\lim _{x \rightarrow c} f(x)$ exists, can you find its value by calculating $\lim _{x \rightarrow c^{+}} f(x) ?$ Give reasons for your answer.
Matt J.
Numerade Educator
Problem 49
Suppose that $f$ is an odd function of $x .$ Does knowing that $\lim _{x \rightarrow 0^{+}} f(x)=3$ tell you anything about lim $_{x \rightarrow 0} f(x) ?$ Give reasons for your answer.
Problem 50
Suppose that $f$ is an even function of $x .$ Does knowing that $\lim _{x \rightarrow 2^{-}} f(x)=7$ tell you anything about either $\lim _{x \rightarrow-2^{-}} f(x)$ or $\lim _{x \rightarrow-2^{+}} f(x) ?$ Give reasons for your answer.
Matt J.
Numerade Educator
Problem 51
Given $\varepsilon>0,$ find an interval $I=(5,5+\delta), \delta>0,$ such that if $x$ lies in $I,$ then $\sqrt{x-5}<\varepsilon$ . What limit is being verified and what is its value?
Problem 52
Given $\varepsilon>0,$ find an interval $I=(4-\delta, 4), \delta>0,$ such that if $x$ lies in $I,$ then $\sqrt{4-x}<\varepsilon$ . What limit is being verified and what is its value?
Matt J.
Numerade Educator
Problem 53
Use the definitions of right-hand and left-hand limits to prove the limit statements.
\begin{equation}\lim _{x \rightarrow 0} \frac{x}{|x|}=-1\end{equation}
Problem 54
Use the definitions of right-hand and left-hand limits to prove the limit statements.
\begin{equation}\lim _{x \rightarrow 2^{+}} \frac{x-2}{|x-2|}=1\end{equation}
Matt J.
Numerade Educator
Problem 55
Greatest integer function Find (a) $\lim _{x \rightarrow 400^{+}}\lfloor x\rfloor$ and (b) $\lim _{x \rightarrow 400^{-}}\lfloor x\rfloor ;$ then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about lim $_{x \rightarrow 400}\lfloor x\rfloor ?$ Give reasons for your answer.
Problem 56
One-sided limits
\begin{equation}f(x)=\left\{\begin{array}{ll}{x^{2} \sin (1 / x),} & {x<0} \\ {\sqrt{x},} & {x>0}.\end{array}\right.\end{equation}
Find (a) $\lim _{x \rightarrow 0^{+}} f(x)$ and (b) $\lim _{x \rightarrow 0 ^{-}} f(x) ;$ then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about $\lim _{x \rightarrow 0} f(x) ?$ Give reasons for your answer.
Matt J.
Numerade Educator
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