4.7E: Exercises for Section 4.7
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- Jun 30, 2021
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In exercises 1 - 6, evaluate the limit.
1) Evaluate the limit limx→∞exx.
2) Evaluate the limit limx→∞exxk.
- Answer
- limx→∞exxk=∞
3) Evaluate the limit limx→∞lnxxk.
4) Evaluate the limit limx→ax−ax2−a2.
- Answer
- limx→ax−ax2−a2=12a
5. Evaluate the limit limx→ax−ax3−a3.
6. Evaluate the limit limx→ax−axn−an.
- Answer
- limx→ax−axn−an=1nan−1
In exercises 7 - 11, 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.
7) limx→0+x2lnx
8) limx→∞x1/x
- Answer
- Cannot apply directly; use logarithms
9) limx→0x2/x
10) limx→0x21/x
- Answer
- Cannot apply directly; rewrite as limx→0x3
11) limx→∞exx
In exercises 12 - 40, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
12) limx→3x2−9x−3
- Answer
- limx→3x2−9x−3=6
13) limx→3x2−9x+3
14) limx→0(1+x)−2−1x
- Answer
- limx→0(1+x)−2−1x=−2
15) \displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}
16) \displaystyle \lim_{x→π}\frac{x−π}{\sin x}
- Answer
- \displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1
17) \displaystyle \lim_{x→1}\frac{x−1}{\sin x}
18) \displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}
- Answer
- \displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n
19) \displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}
20) \displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}
- Answer
- \displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}
21) \displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}
22) \displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}
- Answer
- \displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}
23) \displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}
24) \displaystyle \lim_{x→1}\frac{x-1}{\ln x}
- Answer
- \displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1
25) \displaystyle \lim_{x→0}\,(x+1)^{1/x}
26) \displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}
- Answer
- \displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}
27) \displaystyle \lim_{x→0^+}x^{2x}
28) \displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)
- Answer
- \displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1
29) \displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}
30) \displaystyle \lim_{x→0^+}x\ln\left(x^4\right)
- Answer
- \displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0
31) \displaystyle \lim_{x→∞}(x−e^x)
32) \displaystyle \lim_{x→∞}x^2e^{−x}
- Answer
- \displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0
33) \displaystyle \lim_{x→0}\frac{3^x−2^x}{x}
34) \displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}
- Answer
- \displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1
35) \displaystyle \lim_{x→π/4}(1−\tan x)\cot x
36) \displaystyle \lim_{x→∞}xe^{1/x}
- Answer
- \displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞
37) \displaystyle \lim_{x→0}x^{1/\cos x}
38) \displaystyle \lim_{x→0^{+} }x^{1/x}
- Answer
- \displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0
39) \displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x
40) \displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x
- Answer
- \displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}
For exercises 41 - 50, 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.
41) [T] \displaystyle \lim_{x→0}\frac{e^x−1}{x}
42) [T] \displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)
- Answer
- \displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0
43) [T] \displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}
44) [T] \displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}
- Answer
- \displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1
45) [T] \displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}
46) [T] \displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}
- Answer
- \displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0
47) [T] \displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)
48) [T] \displaystyle \lim_{x→0^+}\tan\left(x^x\right)
- Answer
- \displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1
49) [T] \displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}
50) [T] \displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}
- Answer
- \displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2