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Limits and L'Hopital's Rule

Find the limits using L'Hopital's rule where appropriate. If there is a more elementary method, consider using it. If L'Hospital's rule does not apply explain why.

1) lim as x approaches -1 (x^2 -1) / (x + 1)
2) lim as x approaches -1 (x^9 -1) / (x^5 - 1)
3) lim as x approaches -2 (x+2) / (x^2 +3x + 2)
4) lim as x approaches 0 (x + tanx) / (sinx)
5) lim as x approaches 0 (e^t -1) / (t^3)
6) lim as x approaches 0 (e^3t -1) / (t)
7) lim as x approaches ∞ (ln x) / (x)
8) lim as x approaches ∞ (e^x) / (x)
9) lim as x approaches 0+ (ln x) / (x)
10) lim as x approaches ∞ (ln ln x) / (x)
11) lim as x approaches 0 (5^t -3^t) / (t)
12) lim as x approaches -1 (ln x) / (sin pi x)
13) lim as x approaches 0 (e^x -1 -x) / (x^2)
14) lim as x approaches 0 (e^x - 1 - x - (x^2 / 2)) / (x^3)
15) lim as x approaches ∞ (e^x) / (x^3)
16) lim as x approaches 0 (sinx) / (sinh x)
17) lim as x approaches 0 (sin^-1 x) / (x)
18) lim as x approaches 0 (sinx - x) / (x^3)
19) lim as x approaches 0 (1-cosx) / (x^2)
20) lim as x approaches ∞ (ln x)^2 / (x)
21) lim as x approaches 0 (x + sinx) / (x + cosx)
22) lim as x approaches ∞ (x) / (ln(1 + 2e^x))
23) lim as x approaches 0 (x) / (tan^-1(4x))
24) lim as x approaches -1 (1-x+lnx) / (1+ cos pix)
25) lim as x approaches ∞ (SQRT(x^2 +2)) / (SQRT(2x^2 + 1))
26) lim as x approaches 0 (1-e^(-2x)) / (sec x)
27) lim as x approaches 0+ (SQRT(x) ln x)
28) lim as x approaches -∞ (x^2 * e^x)
29) lim as x approaches 0 (cot 2x sin 6x)
30) lim as x approaches 0+ (sin x ln x)
31) lim as x approaches ∞ (x^3 e^(-x^2)
32) lim as x approaches 1+ (ln x tan(pix/2))
33) lim as x approaches 0 (1-2x)^(1/x)
34) lim as x approaches ∞ [(x) / (x + 1)]^x
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Solution Summary

Thirty-four limit problems are solved (many using L'Hopital's Rule). The solution is 19 pages long and explain in details how to use L'hopital rules.

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