4.8E: Exercises
- Page ID
- 10859
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
For the following exercises, evaluate the limit.
1) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\).
2) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k}\).
- Answer
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\(∞\)
3) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{lnx}{x^k}\).
4) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).
- Answer
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\(\frac{1}{2a}\)
5) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}\).
6) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}\).
- Answer
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\(\frac{1}{na^{n−1}}\)
Exercise \(\PageIndex{2}\)
For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
1) \(\displaystyle \lim_{x→0^+}x^2lnx\)
2) \(\displaystyle \lim_{x→∞}x^{1/x}\)
- Answer
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Cannot apply directly; use logarithms
3) \(\displaystyle \lim_{x→0}x^{2/x}\)
4) \(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)
- Answer
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Cannot apply directly; rewrite as \(\displaystyle \lim_{x→0}x^3\)
5) \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)
Exercise \(\PageIndex{3}\)
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
1) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)
- Answer
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\(6\)
2) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)
3) \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)
- Answer
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\(−2\)
4) \(\displaystyle \lim_{x→π/2}\frac{cosx}{\frac{π}{2}−x}\)
5) \(\displaystyle \lim_{x→π}\frac{x−π}{sinx}\)
- Answer
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\(−1\)
6) \(\displaystyle \lim_{x→1}\frac{x−1}{sinx}\)
7) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}\)
- Answer
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\(n\)
8) \(\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}\)
9) \(\displaystyle \lim_{x→0}\frac{sinx−tanx}{x^3}\)
- Answer
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\(−\frac{1}{2}\)
- Solution
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\(\displaystyle \lim_{x→0}\frac{sinx−tanx}{x^3}=\frac{sin(0)−tan(0)}{0^3}=\left[\dfrac{0}{0}\right].\) By L'hopital's rule,
\(\displaystyle \lim_{x→0}\frac{sinx−tanx}{x^3}=\lim_{x→0}\frac{cos(x)-sec^2(x)}{3x^2}=\lim_{x→0}\frac{cos^3(x)-1}{3x^2 \cos^2(x)}=\left[\dfrac{0}{0}\right].\) By L'hopital's rule,
\(\displaystyle\lim_{x→0}\frac{cos^3(x)-1}{3x^2 \cos^2(x)}=\lim_{x→0}\frac{-3cos^2(x)sin(x)}{3\left(2x cos^2(x)-2x^2 cos(x) sin(x)\right)}= \lim_{x→0}\frac{-cos(x)sin(x)}{\left(2x cos(x)-2x^2 sin(x)\right)}=\left[\dfrac{0}{0}\right].\) By L'hopital's rule,
\(\displaystyle \lim_{x→0}\frac{-cos(x)sin(x)}{\left(2x cos(x)-2x^2 sin(x)\right)}= \lim_{x→0}\frac{\left(-cos(x)cos(x)+sin(x)sin(x)\right)}{\left(2 cos(x) -2x sin(x) \right)-2 \left(2x sin(x)+x^2 cos (x)\right)}= \dfrac{-1^2}{2}= −\frac{1}{2}\)
Hence, \(\displaystyle \lim_{x→0}\frac{sinx−tanx}{x^3}= −\frac{1}{2}\).
10) \(\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}\)
11) \(\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}\)
- Answer
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\(\frac{1}{2}\)
12) \(\displaystyle \lim_{x→0}\frac{tanx}{\sqrt{x}}\)
13) \(\displaystyle \lim_{x→1}\frac{x→1}{lnx}\)
- Answer
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\(1\)
14) \(\displaystyle \lim_{x→0}(x+1)^{1/x}\)
15) \(\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}\)
- Answer
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\(\frac{1}{6}\)
16) \(\displaystyle \lim_{x→0^+}x^{2x}\)
17) \(\displaystyle \lim_{x→∞}xsin(\frac{1}{x})\)
- Answer
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\(1\)
18) \(\displaystyle \lim_{x→0}\frac{sinx−x}{x^2}\)
19) \(\displaystyle \lim_{x→0^+}xln(x^4)\)
- Answer
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\(0\)
20) \(\displaystyle \lim_{x→∞}(x−e^x)\)
21) \(\displaystyle \lim_{x→∞}x^2e^{−x}\)
- Answer
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\(0\)
22) \(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)
23) \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)
- Answer
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\(−1\)
24) \(\displaystyle \lim_{x→π/4}(1−tanx)cotx\)
25) \(\displaystyle \lim_{x→∞}xe^{1/}\)x
- Answer
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\(∞\)
26) \(\displaystyle \lim_{x→0}x^{1/cosx}\)
27) \(\displaystyle \lim_{x→0}x^{1/x}\)
- Answer
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\(1\)
28) \(\displaystyle \lim_{x→0}(1−\frac{1}{x})^x\)
29) \(\displaystyle \lim_{x→∞}(1−\frac{1}{x})^x\)
- Answer
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\(\frac{1}{e}\)
Exercise \(\PageIndex{4}\)
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
1) \(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)
2) \(\displaystyle \lim_{x→0}xsin(\frac{1}{x})\)
- Answer
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\(0\)
3) \(\displaystyle \lim_{x→1}\frac{x−1}{1−cos(πx)}\)
4) \(\displaystyle \lim_{x→1}\frac{e^{(x−1)}−1}{x−1}\)
- Answer
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\(1\)
5) \(\displaystyle \lim_{x→1}\frac{(x−1)^2}{lnx}\)
6) \(\displaystyle \lim_{x→π}\frac{1+cosx}{sinx}\)
- Answer
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\(0\)
7) \(\displaystyle \lim_{x→0}(cscx−\frac{1}{x})\)
8) \(\displaystyle \lim_{x→0^+}tan(x^x)\)
- Answer
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\(tan(1)\)
9) \(\displaystyle \lim_{x→0^+}\frac{lnx}{sinx}\)
10) \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)
- Answer
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\(2\)