Limits

Let $\ds f(x)=\left\{ \begin{array}{lll} \frac{x^2-1}{|x-1|}\amp \mbox{if } \amp x\not= 1,\\ 4\amp \mbox{if } \amp x= 1. \end{array} \right.$ Find $\displaystyle \lim _{x\to 1^-}f(x)\text{.}$

Hint

Which statement is true for $x\lt 1\text{:}$ $|x-1|=x-1$ or $|x-1|=1-x\text{?}$

Answer

$-2\text{.}$

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21.

Let $F(x)=\frac{2x^2-3x}{|2x-3|}\text{.}$

Find $\displaystyle \lim _{x\to 1.5^+}F(x)\text{.}$
Find $\displaystyle \lim _{x\to 1.5^-}F(x)\text{.}$
Does $\displaystyle \lim _{x\to 1.5}F(x)$ exist? Provide a reason.

Answer

$1.5\text{.}$
$-1.5\text{.}$
No. The left-hand limit and the right-hand limit are not equal.

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Evaluate the following limits. If any of them fail to exist, say so and say why.

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22.

$\displaystyle \lim _{x\to -2} \frac{2-|x|}{2+x}$

Answer

$1\text{.}$

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23.

$\displaystyle \lim _{x\to 2^-} \frac{|x^2-4|}{10-5x}$

Answer

$\ds \frac{4}{5}\text{.}$

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24.

$\displaystyle \lim _{x\to 4^-} \frac{|x-4|}{(x-4)^2}$

Answer

$\infty\text{.}$

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25.

$\displaystyle \lim _{x\to 8} \frac{(x-8)(x+2)}{|x-8|}$

Answer

Does not exist.

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26.

$\displaystyle \lim _{x\to 2} \left(\frac{1}{x^2+5x+6}-\frac{1}{x-2}\right)$

Answer

Does not exist.

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27.

$\displaystyle \lim _{x\to -1} \frac{x^2-x-2}{3x^2-x-1}$

Answer

$0\text{.}$

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28.

$\displaystyle \lim _{x\to 16}\frac{\sqrt{x}-4}{x-16}$

Hint

Rationalize the numerator.

Answer

$\displaystyle \frac{1}{8}\text{.}$

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29.

$\displaystyle \lim _{x\to 8}\frac{\sqrt[3]{x}-2}{x-8}$

Hint

Note that $x-8=(\sqrt[3]{x}-2)(\sqrt[3]{x^2}+2\sqrt[3]{x}+4)\text{.}$

Answer

$\displaystyle \frac{1}{12}\text{.}$

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30.

$\displaystyle \lim _{x\to 4} \frac{2-\sqrt{x}}{4x-x^2}$

Answer

$\ds \frac{1}{16}$

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31.

$\displaystyle \lim _{x\to 0} \frac{\sqrt{1+2x}-\sqrt{1-4x}}{x}$

Answer

$3\text{.}$

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32.

Find constants $a$ and $b$ such that $\displaystyle \lim _{x\to 0}\frac{\sqrt{ax+b}-2}{x}=1\text{.}$

Hint

Rationalize the numerator. Choose the value of $b$ so that $x$ becomes a factor in the numerator.

Answer

$a=b=4\text{.}$

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Evaluate the following limits. If any of them fail to exist, say so and say why.

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33.

$\displaystyle \lim _{x\to 5}e^{ \frac{x-5}{\sqrt{x-1}-2}}$

Answer

$\ds e^4\text{.}$

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34.

$\displaystyle \lim _{x\to 7}e^{ \frac{\sqrt{x+2}-3}{x-7}}$

Answer

$\ds e^{1/6}\text{.}$

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35.

$\displaystyle \lim _{t\to 0} \frac{\sqrt{\sin t +1}-1}{t}$

Answer

$\ds \frac{1}{2}\text{.}$

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36.

$\displaystyle \lim _{x\to 8}\frac{x^{1/3}-2}{x-8}$

Hint

Note that $x-8=(\sqrt[3]{x}-2)(\sqrt[3]{x^2}+2\sqrt[3]{x}+4)\text{.}$

Answer

$\displaystyle\frac{1}{12}\text{.}$

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37.

$\displaystyle \lim _{x\to \infty }\left( \sqrt{x^2+x}-x\right)$

Hint

Rationalize the numerator.

Answer

$\displaystyle \frac{1}{2}\text{.}$

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38.

$\displaystyle \lim _{x\to -\infty }\left( \sqrt{x^2+5x}-\sqrt{x^2+2x}\right)$

Hint

Rationalize the numerator. Note that $x\to -\infty$ and use the fact that if $x\lt 0$ then $x=-\sqrt{x^2}\text{.}$

Answer

$\displaystyle -\frac{3}{2}\text{.}$

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39.

$\displaystyle \lim _{x\to \infty }\left( \sqrt{x^2-x+1}-\sqrt{x^2+1}\right)$

Answer

$\displaystyle -\frac{1}{2}\text{.}$

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40.

$\displaystyle \lim _{x\to \infty }\left( \sqrt{x^2+3x-2}-x\right)$

Answer

$\displaystyle \frac{3}{2}\text{.}$

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41.

Is there a number $b$ such that $\ds \lim _{x\to -2}\frac{bx^2+15x+15+b}{x^2+x-2}$ exists? If so, find the value of $b$ and the value of the limit.

Answer

$b=3\text{.}$

Solution

Since the denominator approaches $0$ as $x\to -2\text{,}$ the necessary condition for this limit to exist is that the numerator approaches $0$ as $x\to -2\text{.}$ Thus we solve $4b-30+15+b=0$ to obtain $b=3\text{.}$ $\ds \lim _{x\to -2}\frac{3x^2+15x+18}{x^2+x-2}=-1\text{.}$

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42.

Determine the value of $a$ so that $\ds f(x)=\frac{x^2+ax+5}{x+1}$ has a slant asymptote $y=x+3\text{.}$

Hint

Write $\ds f(x)=x+\frac{(a-1)x+5}{x+1}\text{.}$

Answer

$a=4$

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43.

Prove that $f(x)=\frac{\ln x}{x}$ has a horizontal asymptote $y=0\text{.}$

Answer

$\displaystyle \lim _{x\to \infty}\frac{\ln x}{x}=0\text{.}$

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44.

Let $I$ be an open interval such that $4\in I$ and let a function $f$ be defined on a set $D=I\backslash \{ 4\}\text{.}$ Evaluate $\displaystyle \lim _{x\to 4}f(x)\text{,}$ where $x+2\leq f(x)\leq x^2-10$ for all $x\in D\text{.}$

Answer

Solution

From $\displaystyle \lim _{x\to 4}(x+2)=6$ and $\displaystyle \lim _{x\to 4}(x^2-10)=6\text{,}$ by the Squeeze Theorem, it follows that $\displaystyle \lim _{x\to 4}f(x)=6\text{.}$

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45.

Evaluate $\displaystyle \lim _{x\to 1}f(x)\text{,}$ where $2x-1\leq f(x)\leq x^2$ for all $x$ in the interval $(0,2)\text{.}$

Answer

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Use the squeeze theorem to show that

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46.

$\displaystyle \lim _{x\to 0}x^4\sin\left( \frac{1}{x}\right) =0\text{.}$

Solution

Use the fact $\ds -x^4\leq x^4\sin\left(\frac{1}{x}\right)\leq x^4\text{,}$ $x\not= 0\text{.}$

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47.

$\displaystyle \lim _{x\to 0^+}\left( \sqrt{x}e^{\sin (1/x)}\right) =0\text{.}$

Solution

From the fact that $\displaystyle \left| \sin (1/x)\right|\leq 1$ for all $x\not= 0$ and the fact that the function $\displaystyle y=e^x$ is increasing conclude that $\displaystyle e^{-1}\leq e^{\sin (1/x)}\leq e$ for all $x\not= 0\text{.}$ Thus $\displaystyle e^{-1} \cdot \sqrt{x} \leq \sqrt{x}e^{\sin (1/x)} \leq e\cdot \sqrt{x}$ for all $x>0\text{.}$ By the Squeeze Theorem, $\displaystyle \lim _{x\to 0^+}\left( \sqrt{x}e^{\sin (1/x)}\right) =0\text{.}$

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Evaluate the following limits. If any of them fail to exist, say so and say why.

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48.

$\displaystyle \lim _{x\to 0^+}\left[ (x^2+x)^{1/3}\sin \left( \frac{1}{x^2}\right)\right]$

Hint

Squeeze Theorem.

Answer

$0\text{.}$

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49.

$\displaystyle \lim _{x\to 0}x\sin \left( \frac{e}{x}\right)$

Hint

Squeeze Theorem.

Answer

$0\text{.}$

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50.

$\displaystyle \lim _{x\to 0}x\sin \left( \frac{1}{x^2}\right)$

Hint

Squeeze Theorem.

Answer

$0\text{.}$

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51.

$\displaystyle \lim _{x\to 0} \sqrt{x^2+x}\cdot \sin\left(\frac{\pi}{x}\right)$

Hint

Squeeze Theorem

Answer

$0\text{.}$

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52.

$\displaystyle \lim _{x\to 0} x\cos^2\left(\frac{1}{x^2}\right)$

Hint

Squeeze Theorem.

Answer

$0\text{.}$

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53.

$\displaystyle \lim _{x\to \pi /2^+}\frac{x}{\cot x}$

Answer

$-\infty\text{.}$

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54.

$\displaystyle \lim _{x\to 0}\frac{1-e^{-x}}{1-x}$

Answer

$0\text{.}$

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55.

$\displaystyle \lim _{x\to 0} \frac{e^{2x}-1-2x}{x^2}$

Answer

$2\text{.}$

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56.

$\displaystyle \lim _{x\to 2} \frac{e^x-e^2}{\cos\left(\frac{\pi x}{2}\right)+1}$

Answer

Does not exist.

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57.

$\displaystyle \lim _{x\to 1} \frac{x^2-1}{e^{1-x^7}-1}$

Answer

$\ds -\frac{2}{7}\text{.}$

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58.

$\displaystyle \lim _{x\to 0}\frac{e^{-x^2}\cos (x^2)}{x^2}$

Answer

$\infty\text{.}$

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59.

$\displaystyle \lim _{x\to 1}\frac{x^{76}-1}{x^{45}-1}$

Hint

This is the case “$0/0$”. Apply L'Hôpital's rule.

Answer

$\displaystyle \frac{76}{45}\text{.}$

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60.

$\displaystyle \lim _{x\to 1} \frac{x^a-1}{x^b-1}\text{,}$ $a,b\not=0$

Answer

$\ds \frac{a}{b}\text{.}$

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61.

$\displaystyle \lim _{x\to 0}\frac{(\sin x)^{100}}{x^{99}\sin 2x}$

Hint

Write $\displaystyle \frac{1}{2}\cdot \left( \frac{\sin x}{x}\right) ^{100}\cdot \frac{2x}{\sin 2x}\text{.}$

Answer

$\displaystyle \frac{1}{2}\text{.}$

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62.

$\displaystyle \lim _{x\to 0}\frac{x^{100}\sin 7x}{(\sin x)^{99}}$

Hint

Write $\displaystyle 7\cdot \left( \frac{x}{\sin x}\right) ^{101}\cdot \frac{\sin 7x}{7x}\text{.}$

Answer

$7. $

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63.

$\displaystyle \lim _{x\to 0}\frac{x^{100}\sin 7x}{(\sin x)^{101}}$

Answer

$7. $

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64.

$\displaystyle \lim _{x\to 0}\frac{\arcsin 3x}{\arcsin 5x}$

Hint

This is the case “$0/0$”. Apply L'Hôpital's rule.

Answer

$\displaystyle \frac{3}{5}\text{.}$

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65.

$\displaystyle \lim _{x\to 0}\frac{\sin 3x}{\sin 5x}$

Answer

$\displaystyle \frac{3}{5}\text{.}$

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66.

$\displaystyle \lim _{x\to 0} \frac{x^3\sin \left( \frac{1}{x^2}\right)}{\sin x}$

Hint

Write $\displaystyle x^2 \cdot \frac{x}{\sin x}\cdot \sin \left( \frac{1}{x^2}\right)\text{.}$

Answer

$0\text{.}$

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67.

$\displaystyle \lim _{x\to 0}\frac{\sin x}{\sqrt{x\sin 4x}}$

Hint

$\displaystyle \frac{\sin x}{2|x|}\cdot \frac{1}{\sqrt{\frac{\sin 4x}{4x}}}\text{.}$

Answer

Does not exist.

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68.

$\displaystyle \lim _{x \to 0}\frac{1-\cos x}{x\sin x}$

Hint

Write $\displaystyle \frac{1-\cos x}{x^2}\cdot\frac{x}{\sin x}\text{.}$

Answer

$\displaystyle \frac{1}{2}\text{.}$

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69.

$\displaystyle \lim _{\theta\to \frac{3\pi}{2}} \frac{\cos \theta +1}{\sin\theta}$

Answer

$-1\text{.}$

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70.

$\displaystyle \lim _{x\to \frac{\pi}{2}} \left(x-\frac{\pi}{2}\right)\tan x$

Answer

$-1\text{.}$

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71.

$\displaystyle \lim _{x\to \infty }x\tan (1/x)$

Hint

Substitute $\displaystyle t=\frac{1}{x}\text{.}$

Answer

$1\text{.}$

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72.

$\displaystyle \lim _{x\to 0}\left( \frac{1}{\sin x}-\frac{1}{x}\right)$

Hint

This is the case $"\infty - \infty"\text{.}$ Write $\displaystyle \frac{x- \sin x}{x\sin x}$ and apply L'Hôpital's rule.

Answer

$0\text{.}$

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73.

$\displaystyle \lim _{x\to 0}\frac{x- \sin x}{x^3}$

Answer

$\ds \frac{1}{6}.$

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74.

$\displaystyle \lim _{x\to 0} (\csc x-\cot x)$

Answer

$0\text{.}$

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75.

$\displaystyle \lim _{x\to 0^+}(\sin x)(\ln \sin x)$

Hint

This is the case $``0\cdot \infty ''\text{.}$ Write $\displaystyle \frac{\ln \sin x}{\frac{1}{\sin x}}$ and apply L'Hôpital's rule.

Answer

$0\text{.}$

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76.

$\displaystyle \lim _{x\to \infty} \left(x\cdot \ln\frac{x-1}{x+1}\right)$

Answer

$-2\text{.}$

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77.

$\displaystyle \lim _{x\to \infty} \frac{e^{\frac{x}{10}}}{x^3}$

Answer

$\infty\text{.}$

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78.

$\displaystyle \lim _{x\to \infty }\frac{\ln x}{\sqrt{x}}$

Hint

This is the case $``\infty /\infty ''\text{.}$ Apply L'Hôpital's rule.

Answer

$0\text{.}$

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79.

$\displaystyle \lim _{x\to \infty }\frac{\ln 3x}{x^2}$

Answer

$0\text{.}$

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80.

$\displaystyle \lim _{x\to \infty }\frac{(\ln x)^2}{x}$

Answer

$0\text{.}$

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81.

$\displaystyle \lim _{x\to 1 }\frac{\ln x}{x}$

Answer

$0\text{.}$

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82.

$\displaystyle \lim _{x\to 0 }\frac{\ln (2+2x)-\ln 2}{x}$

Hint

This is the case $``0/0''\text{.}$ Write $\displaystyle \frac{\ln (1+x)}{x}$ and apply L'Hôpital's rule.

Answer

$1\text{.}$

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83.

$\displaystyle \lim _{x\to \infty }\frac{\ln ((2x)^{1/2})}{\ln ((3x)^{1/3})}$

Hint

Use properties of logarithms first.

Answer

$\displaystyle \frac{3}{2}\text{.}$

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84.

$\displaystyle \lim _{x\to 0 }\frac{\ln (1+3x)}{2x}$

Answer

$\displaystyle \frac{3}{2}\text{.}$

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85.

$\displaystyle \lim _{x\to 1 }\frac{\ln (1+3x)}{2x}$

Hint

The denominator approaches 2.

Answer

$\ln 2\text{.}$

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86.

$\displaystyle \lim _{\theta \to \frac{\pi }{2} ^+}\frac{\ln (\sin \theta)}{\cos \theta }$

Hint

This is the case “$0/0$”. Apply L'Hospital's rule.

Answer

$0\text{.}$

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87.

$\displaystyle \lim _{x\to 1 }\frac{1-x+\ln x}{1+\cos (\pi x)}$

Hint

Apply L'Hospital's rule twice.

Answer

$\displaystyle -\frac{1}{\pi ^2}\text{.}$

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88.

$\displaystyle \lim _{x\to 0 }\left( \frac{1}{x^2}-\frac{1}{\tan x}\right)$

Hint

This is the case "$\infty - \infty$". Write $\displaystyle \frac{\sin x-x^2\cos x}{x^2\sin x}$ and apply L'Hospital's rule.

Answer

$\infty\text{.}$

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89.

$\displaystyle \lim _{x\to 0^+} \left(\frac{1}{x}-\frac{1}{e^x-1}\right)$

Answer

$\ds \frac{1}{2}\text{.}$

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90.

$\displaystyle\lim _{x\to 0}(\cosh x)^{\frac{1}{x^2}}$

Hint

This is the case "$\displaystyle 1^\infty$". Write $\displaystyle e^{\frac{\ln \cosh x}{x^2}}\text{.}$ Apply L'Hospital's rule and use the fact that the exponential function $f(x)=e^x$ is continuous.

Answer

$\displaystyle e^{\frac{1}{2}}\text{.}$

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91.

$\displaystyle \lim _{x\to 0^+}(\cos x)^{\frac{1}{x}}$

Answer

$1\text{.}$

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92.

$\displaystyle \lim _{x\to 0^+}(\cos x)^{\frac{1}{x^2}}$

Answer

$\ds e^{-1/2}\text{.}$

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93.

$\displaystyle \lim _{x\to 0^+}x^{x}$

Hint

This is the case "$\displaystyle 0^0$". Write $\displaystyle x^x=e^{x\ln x}=e^{\frac{\ln x}{x^{-1}}}\text{.}$ Apply L'Hospital's rule and use the fact that the exponential function $f(x)=e^x$ is continuous.

Answer

$1\text{.}$

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94.

$\displaystyle \lim _{x\to 0^+}x^{\sqrt{x}}$

Answer

$1\text{.}$

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95.

$\displaystyle \lim _{x\to 0^+} x^{\tan x}$

Answer

$1\text{.}$

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96.

$\displaystyle \lim _{x\to 0^+}(\sin x)^{\tan x}$

Answer

$1\text{.}$

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97.

$\displaystyle \lim _{x\to 0}(1+\sin x)^{\frac{1}{x}}$

Answer

$e\text{.}$

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98.

$\displaystyle \lim _{x\to \infty }(x+\sin x)^{\frac{1}{x}}$

Hint

This is the case "$\infty^0$".

Answer

$1\text{.}$

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99.

$\displaystyle \lim _{x\to \infty }x^{\frac{1}{x}}$

Answer

$1\text{.}$

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100.

$\displaystyle \lim _{x\to \infty }\left( 1+ \frac{1}{x}\right) ^{2x}$

Answer

$e^2\text{.}$

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101.

$\displaystyle \lim _{x\to \infty }\left( 1+\sin \frac{3}{x}\right) ^x$

Answer

$\displaystyle e^3\text{.}$

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102.

$\displaystyle \lim _{x\to 0^+}(x+\sin x)^{\frac{1}{x}}$

Answer

$\displaystyle 0\text{.}$

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103.

$\displaystyle \lim _{x\to 0^+}\left( \frac{x}{x+1}\right) ^{x}$

Hint

Write $\displaystyle e^{x\ln \frac{x}{x+1}}=e^{x\ln x}\cdot e^{-x\ln (x+1)}$ and make your conclusion.

Answer

$1\text{.}$

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104.

$\displaystyle \lim _{x\to e^+}(\ln x)^{\frac{1}{x-e}}$

Answer

$\ds e^{\frac{1}{e}}\text{.}$

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105.

$\displaystyle \lim _{x\to e^+}(\ln x)^{\frac{1}{x}}$

Answer

$1\text{.}$

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106.

$\displaystyle \lim _{x\to 0}e^{x\sin (1/x)}$

Hint

Use the Squeeze Theorem.

Answer

$1\text{.}$

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107.

$\displaystyle \lim _{x\to 0}(1-2x)^{1/x}$

Hint

Write $\displaystyle \left( (1-2x)^{-\frac{1}{2x}}\right) ^{-2}\text{.}$

Answer

$\displaystyle e^{-2}\text{.}$

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108.

$\displaystyle \lim _{x\to 0^+}(1+7x)^{1/5x}$

Hint

Write $\displaystyle \left( (1+7x)^{\frac{1}{7x}}\right) ^{\frac{7}{5}}\text{.}$

Answer

$\displaystyle e^{\frac{7}{5}}\text{.}$

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109.

$\displaystyle \lim _{x\to 0^+}(1+3x)^{1/8x}$

Hint

Write $\displaystyle \left( (1+3x)^{\frac{1}{3x}}\right) ^{\frac{3}{8}}\text{.}$

Answer

$\displaystyle e^{\frac{3}{8}}\text{.}$

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110.

$\displaystyle \lim _{x\to 0}\left( 1+\frac{x}{2}\right) ^{3/x}$

Hint

Write $\displaystyle \left( \left( 1+\frac{x}{2}\right) ^{\frac{2}{x}}\right) ^{\frac{3}{2}}\text{.}$

Answer

$\displaystyle e^{\frac{3}{2}}\text{.}$

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111.

Let $x_1=100\text{,}$ and for $n\geq 1\text{,}$ let $\displaystyle x_{n+1}=\frac{1}{2}\left(x_n+\frac{100}{x_n}\right)\text{.}$ Assume that $\displaystyle L=\lim _{n\to \infty }x_n$ exists. Calculate $L\text{.}$

Hint

Use the fact that $\displaystyle L=\lim _{n\to \infty }x_n$ to conclude $L^2=100\text{.}$ Can $L$ be negative?

Answer

$10\text{.}$

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Compute the following limits, or show that they do not exist.

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112.

$\displaystyle \lim _{x \to 0}\frac{1-\cos x}{x^2}$

Hint

Write $\displaystyle \frac{2\sin ^2 \frac{x}{2}}{x^2}\text{,}$ or use L'Hôpital's rule.

Answer

$\displaystyle \frac{1}{2}\text{.}$

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113.

$\displaystyle\lim _{x \to 2\pi }\frac{1-\cos x}{x^2}\text{.}$

Answer

$0\text{.}$

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114.

$\displaystyle \lim _{x \to -1}\arcsin x\text{.}$

Answer

Does not exist. Note that the domain of $f(x)=\arcsin x$ is the interval $[-1,1]\text{.}$

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Compute the following limits or state why they do not exist:

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115.

$\displaystyle \lim _{h\to 0}\frac{\sqrt[4]{16+h}-2}{2h}$

Answer

$\displaystyle \frac{1}{64}\text{.}$

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116.

$\displaystyle \lim _{x\to 1}\frac{\ln x}{\sin (\pi x)}$

Hint

Use L'Hôpital's rule.

Answer

$\displaystyle -\frac{1}{\pi }\text{.}$

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117.

$\displaystyle \lim _{u\to \infty }\frac{u}{\sqrt{u^2+1}}$

Hint

Divide the numerator and denominator by $u\text{.}$

Answer

$1\text{.}$

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118.

$\displaystyle \lim _{x\to 0 }(1-2x)^{1/x}$

Answer

$e^{-2}\text{.}$

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119.

$\displaystyle \lim _{x\to 0 }\frac{(\sin x)^{100}}{x^{99}\sin (2x)}$

Answer

$\displaystyle \frac{1}{2}$

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120.

$\displaystyle \lim _{x\to \infty }\frac{1.01^x}{x^{100}}$

Hint

Think, exponential vs. polynomial.

Answer

$\infty\text{.}$

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Find the following limits. If a limit does not exist, write 'DNE'. No justification necessary.

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121.

$\displaystyle \lim _{x\to 0}\frac{(2+x)^{2016}-2^{2016}}{x}$

Answer

$2016\cdot 2^{2015}\text{.}$

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122.

$\displaystyle \lim _{x\to \infty }(\sqrt{x^2+x}-x)$

Answer

$\displaystyle \frac{1}{2}\text{.}$

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123.

$\displaystyle \lim _{x\to 0} \cot (3x)\sin (7x)$

Answer

$\displaystyle \frac{7}{3}\text{.}$

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124.

$\displaystyle \lim _{x\to 0^+}x^x$

Answer

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125.

$\displaystyle \lim _{x\to \infty} \frac{x^2}{e^x}$

Answer

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126.

$\displaystyle \lim _{x\to 3}\frac{\sin x-x}{x^3}$

Answer

$\displaystyle \frac{\sin 3 - 3}{27}\text{.}$

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Evaluate the following limits, if they exist.

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127.

$\displaystyle \lim _{x\to 0}\frac{f(x)}{|x|}$ given that $\displaystyle \lim _{x\to 0}xf(x)=3\text{.}$

Hint

Consider $\displaystyle \lim _{x\to 0}\frac{xf(x)}{x|x|}\text{.}$

Answer

Does not exist.

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128.

$\displaystyle \lim _{x\to 1} \frac{\sin (x-1)}{x^2+x-2}$

Answer

$\displaystyle \frac{1}{3}\text{.}$

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129.

$\displaystyle \lim _{x\to -\infty }\frac{\sqrt{x^2+4x}}{4x+1}$

Hint

Note that $x\lt 0\text{.}$

Answer

$\displaystyle -\frac{1}{4}\text{.}$

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130.

$\displaystyle \lim _{x\to \infty }\frac{\sqrt{x^4+2}}{x^4-4}$

Answer

🔗

131.

$\displaystyle \lim _{x\to \infty} (e^x+x)^{1/x}$

Answer

$e\text{.}$

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Evaluate the following limits, if they exist.

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132.

$\displaystyle \lim _{x\to 4}\left[ \frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right]$

Answer

$\ds \frac{1}{4}\text{.}$

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133.

$\displaystyle \lim _{x\to 1} \frac{x^2-1}{e^{1-x^2}-1}$

Answer

$-1\text{.}$

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134.

$\displaystyle \lim _{x\to 0}(\sin x)(\ln x)$

Answer

$0\text{.}$

🔗

Evaluate the following limits. Use “ $\infty$ ” or “ $-\infty$ ” where appropriate.

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135.

$\displaystyle \lim _{x\to 1^-}\frac{x+1}{x^2-1}$

Answer

$-\infty\text{.}$

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136.

$\displaystyle \lim _{x\to 0} \frac{\sin 6x}{2x}$

Answer

$3\text{.}$

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137.

$\displaystyle \lim _{x\to 0}\frac{\sinh 2x}{xe^x}$

Answer

$2\text{.}$

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138.

$\displaystyle \lim _{x\to 0^+}(x^{0.01}\ln x)$

Answer

$0\text{.}$

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139.

Use the $\varepsilon -- \delta$ definition of limits to prove that

$\begin{equation*} \lim _{x\to 0}x^3=0\text{.} \end{equation*}$

Solution

Let $\varepsilon >0$ be given. We need to find $\delta =\delta (\varepsilon )>0$ such that $|x-0|\lt \delta \Rightarrow |x^3-0|\lt \varepsilon\text{,}$ which is the same as $|x|\lt \delta \Rightarrow |x^3|\lt \varepsilon\text{.}$ Clearly, we can take $\delta =\sqrt[3]{\varepsilon }\text{.}$ Indeed, for any $\varepsilon >0$ we have that $|x|\lt \sqrt[3]{\varepsilon } \Rightarrow |x|^3=|x^3|\lt \varepsilon$ and, by definition, $\displaystyle \lim _{x\to 0}x^3=0\text{.}$

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140.

Sketch an approximate graph of $f(x)=2x^2$ on $[0,2]\text{.}$ Next, draw the points $P(1,0)$ and $Q(0,2)\text{.}$ When using the precise definition of $\lim _{x\to 1}f(x)=2\text{,}$ a number $\delta$ and another number $\varepsilon$ are used. Show points on the graph which these values determine. (Recall that the interval determined by $\delta$ must not be greater than a particular interval determined by $\varepsilon\text{.}$ )
Use the graph to find a positive number $\delta$ so that whenever $|x-1|\lt \delta$ it is always true that $|2x^2-2|\lt \frac{1}{4}\text{.}$
State exactly what has to be proved to establish this limit property of the function $f\text{.}$

Answer

For any $\varepsilon >0$ there exists $\delta =\delta (\varepsilon )>0$ such that $|x-1|\lt \delta \Rightarrow |2x^2-2|\lt \varepsilon\text{.}$

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141.

Give an example of a function $F=f+g$ such that the limits of $f$ and $g$ at $a$ do not exist and that the limit of $F$ at $a$ exists.

Answer

Take, for example, $f(x)=\mbox{sign} (x)\text{,}$ $g(x)=-\mbox{sign} (x)\text{,}$ and $a=0\text{.}$

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142.

If $\ds \lim_{x\to a}[f(x)+g(x)]=2$ and $\ds \lim_{x\to a}[f(x)-g(x)]=1$ find $\ds \lim_{x\to a}[f(x)\cdot g(x)]\text{.}$

Answer

$\ds \frac{3}{4}\text{.}$

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143.

If $f'$ is continuous, use L'Hospital's rule to show that

$\begin{equation*} \displaystyle \lim _{h\to 0}\frac{f(x+h)-f(x-h)}{2h}=f'(x)\text{.} \end{equation*}$

Explain the meaning of this equation with the aid of a diagram.

Answer

$\displaystyle \lim _{h\to 0}\frac{f(x+h)-f(x-h)}{2h}=\lim _{h\to 0}\frac{f'(x+h)+f'(x-h)}{2}$ and, since $f'$ is continuous, $\displaystyle \lim _{h\to 0}f'(x+h)=\lim _{h\to 0}f'(x-h)=f'(x)\text{.}$