Skip to main content

Exercises 1.2 Limits

Evaluate the following limits. Use limit theorems, not \(\varepsilon\) - \(\delta\) techniques. If any of them fail to exist, say so and say why.

Evaluate the following limits.

1.

\(\displaystyle \lim _{x\to 10} \frac{x^2-100}{x-10}\)

Answer
\(20\text{.}\)
2.

\(\displaystyle \lim _{x\to 10} \frac{x^2-99}{x-10}\)

Answer
Does not exist.
3.

\(\displaystyle \lim _{x\to 10} \frac{x^2-100}{x-9}\)

Answer
\(0\text{.}\)
4.

\(\ds \lim _{x\to 10} f(x)\text{,}\) where \(f(x)=x^2\) for all \(x\not= 10\text{,}\) but \(f(10)=99\text{.}\)

Answer
\(100\text{.}\)
5.

\(\displaystyle \lim _{x\to 10}\sqrt{-x^2+20x-100}\)

Answer

Does not exist. Consider the domain of \(g(x) =\sqrt{-x^2+20x-100}=\sqrt{-(x-10)^2}\text{.}\)

Evaluate the following limits.

6.

\(\displaystyle \lim _{x\to -4} \frac{x^2-16}{x+4}\ln |x|\)

Answer
\(-8\ln 4\text{.}\)
7.

\(\displaystyle \lim _{x\to \infty} \frac{x^2}{e^{4x}-1-4x}\)

Answer

\(0\text{.}\) Note the exponential function in the denominator.

8.

\(\displaystyle \lim _{x\to -\infty} \frac{3x^6-7x^5+x}{5x^6+4x^5-3}\)

Answer

\(\displaystyle \frac{3}{5}\text{.}\) Divide the numerator and denominator by the highest power.

9.

\(\displaystyle \lim _{x\to -\infty} \frac{5x^7-7x^5+1}{2x^7+6x^6-3}\)

Answer
\(\displaystyle \frac{5}{2}\text{.}\)
10.

\(\displaystyle \lim _{x\to -\infty} \frac{2x+3x^3}{x^3+2x-1}\)

Answer
\(3\text{.}\)
11.

\(\displaystyle \lim _{x\to -\infty} \frac{5x+2x^3}{x^3+x-7}\)

Answer
\(2\text{.}\)
12.

\(\displaystyle \lim _{x\to \infty} \frac{ax^{17}+bx}{cx^{17}-dx^3}\text{,}\) \(a,b,c,d\not=0\)

Answer
\(\ds \frac{a}{c}\text{.}\)
13.

\(\displaystyle \lim _{x\to \infty} \frac{3x+|1-3x|}{1-5x}\)

Hint
What is the value of \(3x+|1-3x|\) if \(x\lt \frac{1}{3}\text{?}\)
Answer
\(0\text{.}\)
14.

\(\displaystyle \lim _{x\to -\infty} \frac{\sqrt{x^6-3}}{\sqrt{x^6+5}}\)

Answer
\(1\text{.}\)
15.

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

Answer
\(1\text{.}\)
16.

\(\displaystyle \lim _{x\to \infty} \frac{1+3x}{\sqrt{2x^2+x}}\)

Answer
\(\displaystyle \frac{3}{\sqrt{2}}\text{.}\)
17.

\(\displaystyle \lim _{x\to \infty} \frac{\sqrt{4x^2+3x}-7}{7-3x}\)

Answer
\(\displaystyle -\frac{2}{3}\text{.}\)
18.

\(\displaystyle \lim _{x\to -\infty} \frac{\sqrt{x^2-9}}{2x-1}\)

Answer
\(\ds -\frac{1}{2}\text{.}\)
19.

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

Hint
Note that \(x^2-1=(x-1)(x+1)\text{.}\)
Answer
\(\infty\text{.}\)
20.

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
\(-2\text{.}\)
21.

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

  1. Find \(\displaystyle \lim _{x\to 1.5^+}F(x)\text{.}\)

  2. Find \(\displaystyle \lim _{x\to 1.5^-}F(x)\text{.}\)

  3. Does \(\displaystyle \lim _{x\to 1.5}F(x)\) exist? Provide a reason.

Answer

  1. \(1.5\text{.}\)

  2. \(-1.5\text{.}\)

  3. No. The left-hand limit and the right-hand limit are not equal.

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

22.

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

Answer
\(1\text{.}\)
23.

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

Answer
\(\ds \frac{4}{5}\text{.}\)
24.

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

Answer
\(\infty\text{.}\)
25.

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

Answer
Does not exist.
26.

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

Answer
Does not exist.
27.

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

Answer
\(0\text{.}\)
28.

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

Hint
Rationalize the numerator.
Answer
\(\displaystyle \frac{1}{8}\text{.}\)
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
\(\displaystyle \frac{1}{12}\text{.}\)
30.

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

Answer
\(\ds \frac{1}{16}\)
31.

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

Answer
\(3\text{.}\)
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
\(a=b=4\text{.}\)

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

33.

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

Answer
\(\ds e^4\text{.}\)
34.

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

Answer
\(\ds e^{1/6}\text{.}\)
35.

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

Answer
\(\ds \frac{1}{2}\text{.}\)
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
\(\displaystyle\frac{1}{12}\text{.}\)
37.

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

Hint
Rationalize the numerator.
Answer
\(\displaystyle \frac{1}{2}\text{.}\)
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
\(\displaystyle -\frac{3}{2}\text{.}\)
39.

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

Answer
\(\displaystyle -\frac{1}{2}\text{.}\)
40.

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

Answer
\(\displaystyle \frac{3}{2}\text{.}\)
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 \(b=3\text{.}\) \(\ds \lim _{x\to -2}\frac{3x^2+15x+18}{x^2+x-2}=-1\text{.}\)

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

\(a=4\)

43.

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

Answer
\(\displaystyle \lim _{x\to \infty}\frac{\ln x}{x}=0\text{.}\)
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
6.
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 \(\displaystyle \lim _{x\to 4}f(x)=6\text{.}\)

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
1.

Use the squeeze theorem to show that

46.

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

Solution

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

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

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

48.

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

Hint
Squeeze Theorem.
Answer
\(0\text{.}\)
49.

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

Hint
Squeeze Theorem.
Answer
\(0\text{.}\)
50.

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

Hint
Squeeze Theorem.
Answer
\(0\text{.}\)
51.

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

Hint
Squeeze Theorem
Answer
\(0\text{.}\)
52.

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

Hint
Squeeze Theorem.
Answer
\(0\text{.}\)
53.

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

Answer
\(-\infty\text{.}\)
54.

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

Answer
\(0\text{.}\)
55.

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

Answer
\(2\text{.}\)
56.

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

Answer
Does not exist.
57.

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

Answer
\(\ds -\frac{2}{7}\text{.}\)
58.

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

Answer
\(\infty\text{.}\)
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
\(\displaystyle \frac{76}{45}\text{.}\)
60.

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

Answer
\(\ds \frac{a}{b}\text{.}\)
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
\(\displaystyle \frac{1}{2}\text{.}\)
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
\(7. \)
63.

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

Answer
\(7. \)
64.

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

Hint
This is the case “\(0/0\)”. Apply L'Hôpital's rule.
Answer
\(\displaystyle \frac{3}{5}\text{.}\)
65.

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

Answer
\(\displaystyle \frac{3}{5}\text{.}\)
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
\(0\text{.}\)
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.
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
\(\displaystyle \frac{1}{2}\text{.}\)
69.

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

Answer
\(-1\text{.}\)
70.

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

Answer
\(-1\text{.}\)
71.

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

Hint
Substitute \(\displaystyle t=\frac{1}{x}\text{.}\)
Answer
\(1\text{.}\)
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
\(0\text{.}\)
73.

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

Answer
\(\ds \frac{1}{6}.\)
74.

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

Answer
\(0\text{.}\)
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
\(0\text{.}\)
76.

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

Answer
\(-2\text{.}\)
77.

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

Answer
\(\infty\text{.}\)
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
\(0\text{.}\)
79.

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

Answer
\(0\text{.}\)
80.

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

Answer
\(0\text{.}\)
81.

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

Answer
\(0\text{.}\)
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
\(1\text{.}\)
83.

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

Hint
Use properties of logarithms first.
Answer
\(\displaystyle \frac{3}{2}\text{.}\)
84.

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

Answer
\(\displaystyle \frac{3}{2}\text{.}\)
85.

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

Hint
The denominator approaches 2.
Answer
\(\ln 2\text{.}\)
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
\(0\text{.}\)
87.

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

Hint
Apply L'Hospital's rule twice.
Answer
\(\displaystyle -\frac{1}{\pi ^2}\text{.}\)
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
\(\infty\text{.}\)
89.

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

Answer
\(\ds \frac{1}{2}\text{.}\)
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
\(\displaystyle e^{\frac{1}{2}}\text{.}\)
91.

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

Answer
\(1\text{.}\)
92.

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

Answer
\(\ds e^{-1/2}\text{.}\)
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
\(1\text{.}\)
94.

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

Answer
\(1\text{.}\)
95.

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

Answer
\(1\text{.}\)
96.

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

Answer
\(1\text{.}\)
97.

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

Answer
\(e\text{.}\)
98.

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

Hint
This is the case "\(\infty^0\)".
Answer
\(1\text{.}\)
99.

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

Answer
\(1\text{.}\)
100.

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

Answer
\(e^2\text{.}\)
101.

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

Answer
\(\displaystyle e^3\text{.}\)
102.

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

Answer
\(\displaystyle 0\text{.}\)
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
\(1\text{.}\)
104.

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

Answer
\(\ds e^{\frac{1}{e}}\text{.}\)
105.

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

Answer
\(1\text{.}\)
106.

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

Hint
Use the Squeeze Theorem.
Answer
\(1\text{.}\)
107.

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

Hint
Write \(\displaystyle \left( (1-2x)^{-\frac{1}{2x}}\right) ^{-2}\text{.}\)
Answer
\(\displaystyle e^{-2}\text{.}\)
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
\(\displaystyle e^{\frac{7}{5}}\text{.}\)
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
\(\displaystyle e^{\frac{3}{8}}\text{.}\)
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
\(\displaystyle e^{\frac{3}{2}}\text{.}\)
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
\(10\text{.}\)
Compute the following limits, or show that they do not exist.
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
\(\displaystyle \frac{1}{2}\text{.}\)
113.

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

Answer
\(0\text{.}\)
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 \([-1,1]\text{.}\)
Compute the following limits or state why they do not exist:
115.

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

Answer
\(\displaystyle \frac{1}{64}\text{.}\)
116.

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

Hint
Use L'Hôpital's rule.
Answer
\(\displaystyle -\frac{1}{\pi }\text{.}\)
117.

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

Hint
Divide the numerator and denominator by \(u\text{.}\)
Answer
\(1\text{.}\)
118.

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

Answer
\(e^{-2}\text{.}\)
119.

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

Answer
\(\displaystyle \frac{1}{2}\)
120.

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

Hint
Think, exponential vs. polynomial.
Answer
\(\infty\text{.}\)
Find the following limits. If a limit does not exist, write 'DNE'. No justification necessary.
121.

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

Answer
\(2016\cdot 2^{2015}\text{.}\)
122.

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

Answer
\(\displaystyle \frac{1}{2}\text{.}\)
123.

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

Answer
\(\displaystyle \frac{7}{3}\text{.}\)
124.

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

Answer
1.
125.

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

Answer
0.
126.

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

Answer
\(\displaystyle \frac{\sin 3 - 3}{27}\text{.}\)

Evaluate the following limits, if they exist.

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.
128.

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

Answer
\(\displaystyle \frac{1}{3}\text{.}\)
129.

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

Hint
Note that \(x\lt 0\text{.}\)
Answer
\(\displaystyle -\frac{1}{4}\text{.}\)
130.

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

Answer
0
131.

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

Answer
\(e\text{.}\)

Evaluate the following limits, if they exist.

132.

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

Answer
\(\ds \frac{1}{4}\text{.}\)
133.

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

Answer
\(-1\text{.}\)
134.

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

Answer
\(0\text{.}\)

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

135.

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

Answer
\(-\infty\text{.}\)
136.

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

Answer
\(3\text{.}\)
137.

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

Answer
\(2\text{.}\)
138.

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

Answer
\(0\text{.}\)
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, \(\displaystyle \lim _{x\to 0}x^3=0\text{.}\)

140.

  1. 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 \(\varepsilon\text{.}\))

  2. Use the graph to find a positive number \(\delta\) so that whenever \(|x-1|\lt \delta\) it is always true that \(|2x^2-2|\lt \frac{1}{4}\text{.}\)

  3. 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 \(|x-1|\lt \delta \Rightarrow |2x^2-2|\lt \varepsilon\text{.}\)
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 \(a=0\text{.}\)

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
\(\ds \frac{3}{4}\text{.}\)
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, \(\displaystyle \lim _{h\to 0}f'(x+h)=\lim _{h\to 0}f'(x-h)=f'(x)\text{.}\)