Problems
For exercises 1 - 6, 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:
- \(\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞\)
3) Evaluate the limit \(\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}\).
4) Evaluate the limit \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}\).
- Answer:
- \(\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)
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:
- \(\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)
For 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) \(\displaystyle \lim_{x→0^+}x^2\ln x\)
8) \(\displaystyle \lim_{x→∞}x^{1/x}\)
- Answer:
- Cannot apply directly; use logarithms
9) \(\displaystyle \lim_{x→0}x^{2/x}\)
10) \(\displaystyle \lim_{x→0}\frac{x^2}{1/x}\)
- Answer:
- Cannot apply directly; rewrite as \(\displaystyle \lim_{x→0}x^3\)
11) \(\displaystyle \lim_{x→∞}\frac{e^x}{x}\)
For exercises 12 - 44, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
12) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}\)
- Answer:
- \(\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6\)
13) \(\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}\)
14) \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}\)
- Answer:
- \(\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)
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) \(\displaystyle \lim_{x\to 1^+} \left[\frac{1}{\ln x}-\frac{1}{1-x}\right]\)
34) \(\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right]\)
- Answer:
- \(\displaystyle \lim_{x\to 3^+} \left[\frac{5}{x^2-9}-\frac{x}{x-3}\right] = \lim_{x\to 3^+} \frac{5-x^2-3x}{x^2-9} \quad = \quad -∞\)
35) \(\displaystyle \lim_{x\to \infty} \frac{\sqrt{2x^2-3}}{x+2}\)
- Note:
- L’Hôpital’s rule fails to help us find this limit, although the form seems appropriate. But you can evaluate this limit using techniques you learned earlier in calculus.
36) \(\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x}\)
- Answer:
- \(\displaystyle \lim_{x\to \infty} \left(\frac{x+7}{x+3}\right)^{x} \quad = \quad e^{4}\)
37) \(\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}\)
38) \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}\)
- Answer:
- \(\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)
39) \(\displaystyle \lim_{x→π/4}(1−\tan x)\cot x\)
40) \(\displaystyle \lim_{x→∞}xe^{1/x}\)
- Answer:
- \(\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞\)
41) \(\displaystyle \lim_{x→0}x^{1/\cos x}\)
42) \(\displaystyle \lim_{x→0^+}x^{1/x}\)
- Answer:
- \(\displaystyle \lim_{x→0^+}x^{1/x} \quad = \quad 0\)
43) \(\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x\)
44) \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x\)
- Answer:
- \(\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)
For exercises 45 - 54, 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.
45) [T] \(\displaystyle \lim_{x→0}\frac{e^x−1}{x}\)
46) [T] \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)\)
- Answer:
- \(\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0\)
47) [T] \(\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}\)
48) [T] \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}\)
- Answer:
- \(\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)
49) [T] \(\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}\)
50) [T] \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}\)
- Answer:
- \(\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0\)
51) [T] \(\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)\)
52) [T] \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right)\)
- Answer:
- \(\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1\)
53) [T] \(\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}\)
54) [T] \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}\)
- Answer:
- \(\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)