# 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.