# 2.7: Chapter 2 Review Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

True or False. In exercises 1 - 4, justify your answer with a proof or a counterexample.

1) A function has to be continuous at $$x=a$$ if the $$\displaystyle \lim_{x→a}f(x)$$ exists.

2) You can use the quotient rule to evaluate $$\displaystyle \lim_{x→0}\frac{\sin x}{x}$$.

False, since we cannot have $$\displaystyle \lim_{x→0}x=0$$ in the denominator.

3) If there is a vertical asymptote at $$x=a$$ for the function $$f(x)$$, then $$f$$ is undefined at the point $$x=a$$.

4) If $$\displaystyle \lim_{x→a}f(x)$$ does not exist, then $$f$$ is undefined at the point $$x=a$$.

False. A jump discontinuity is possible.

5) Using the graph, find each limit or explain why the limit does not exist.

a. $$\displaystyle \lim_{x→−1}f(x)$$

b. $$\displaystyle \lim_{x→1}f(x)$$

c. $$\displaystyle \lim_{x→0^+}f(x)$$

d. $$\displaystyle \lim_{x→2}f(x)$$

In exercises 6 - 15, evaluate the limit algebraically or explain why the limit does not exist.

6) $$\displaystyle \lim_{x→2}\frac{2x^2−3x−2}{x−2}$$

$$5$$

7) $$\displaystyle \lim_{x→0}3x^2−2x+4$$

8) $$\displaystyle \lim_{x→3}\frac{x^3−2x^2−1}{3x−2}$$

$$8/7$$

9) $$\displaystyle \lim_{x→π/2}\frac{\cot x}{\cos x}$$

10) $$\displaystyle \lim_{x→−5}\frac{x^2+25}{x+5}$$

DNE

11) $$\displaystyle \lim_{x→2}\frac{3x^2−2x−8}{x^2−4}$$

12) $$\displaystyle \lim_{x→1}\frac{x^2−1}{x^3−1}$$

$$2/3$$

13) $$\displaystyle \lim_{x→1}\frac{x^2−1}{\sqrt{x}−1}$$

14) $$\displaystyle \lim_{x→4}\frac{4−x}{\sqrt{x}−2}$$

$$−4$$

15) $$\displaystyle \lim_{x→4}\frac{1}{\sqrt{x}−2}$$

In exercises 16 - 17, use the squeeze theorem to prove the limit.

16) $$\displaystyle \lim_{x→0}x^2\cos(2πx)=0$$

Since $$−1≤\cos(2πx)≤1$$, then $$−x^2≤x^2\cos(2πx)≤x^2$$. Since $$\displaystyle \lim_{x→0}x^2=0=\lim_{x→0}−x^2$$, it follows that $$\displaystyle \lim_{x→0}x^2\cos(2πx)=0$$.

17) $$\displaystyle \lim_{x→0}x^3\sin\left(\frac{π}{x}\right)=0$$

18) Determine the domain such that the function $$f(x)=\sqrt{x−2}+xe^x$$ is continuous over its domain.

$$[2,∞]$$

In exercises 19 - 20, determine the value of $$c$$ such that the function remains continuous. Draw your resulting function to ensure it is continuous.

19) $$f(x)=\begin{cases}x^2+1, & \text{if } x>c\\2^x, & \text{if } x≤c\end{cases}$$

20) $$f(x)=\begin{cases}\sqrt{x+1}, & \text{if } x>−1\\x^2+c, & \text{if } x≤−1\end{cases}$$

In exercises 21 - 22, use the precise definition of limit to prove the limit.

21) $$\displaystyle \lim_{x→1}\,(8x+16)=24$$

22) $$\displaystyle \lim_{x→0}x^3=0$$

$$δ=\sqrt[3]{ε}$$

23) A ball is thrown into the air and the vertical position is given by $$x(t)=−4.9t^2+25t+5$$. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

24) A particle moving along a line has a displacement according to the function $$x(t)=t^2−2t+4$$, where $$x$$ is measured in meters and $$t$$ is measured in seconds. Find the average velocity over the time period $$t=[0,2]$$.

$$0$$ m/sec
25) From the previous exercises, estimate the instantaneous velocity at $$t=2$$ by checking the average velocity within $$t=0.01$$ sec.