1. \( 0 \)
2. \( -\frac{1}{2} \)
3. \( -1 \)
4. \( \frac{1}{2} \)
5. \( \frac{1}{2} \)
6. \( 2 \)
7. \( \frac{\pi^2}{2} \)
8. \( 2 \)
9. \( 0 \)
10. \( 1 \)
11. \( \frac{\ln^2 10 – \ln^2 5}{2} \)
12. \( 0 \)
13. \( 1 \)
14. \( \frac{1}{2} \)
15. \( -\frac{1}{4} \)
16. \( \frac{1}{12} \)
17. \( \frac{1}{2} \)
18. \( -1 \)
19. \( \frac{1}{2} \)
20. \( \frac{1}{3} \)
21. \( \frac{1}{6} \)
22. \( 0 \)
23. \( 1 \)
24. \( \frac{2}{\pi} \)
25. \( -\frac{4a^2}{\pi} \)
26. \( 1 \)
27. \( 1 \)
28. \( \frac{1}{e} \)
29. \( e^{-2} \)
30. \( 1 \)
31. \( 1 \)
32. \( 1 \)
33. \( 1 \)
34. \( \frac{1}{e} \)
35. \( 1 \)
36. \( \frac{2}{3} \)
37. \( +\infty \)
38. \( e^2 \)
39. \( \frac{1}{2} \)
40. \( e \)
41. \( 0 \)
42. \( -8 \)
43. \( 0 \)

En los problemas del 1 al 43, hallar el límite indicado.

1. \[ \lim_{x\to a} \frac{x^3 – ax^2 – a^2x + a^3}{x^2 – a^2} \]
2. \[ \lim_{x\to 0} \frac{x – e^x + 1}{x^2} \]
3. \[ \lim_{x\to \pi} \frac{\operatorname{sen} x}{x – \pi} \]
4. \[ \lim_{x\to \pi} \frac{1 + \cos x}{\tan^2 x} \]
5. \[ \lim_{x\to \frac{\pi}{4}} \frac{\sec^2 x – 2\tan x}{1 + \cos 4x} \]
6. \[ \lim_{x\to 0^-} \frac{\cot x}{\cot 2x} \]
7. \[ \lim_{x\to 0^-} \frac{\frac{\pi}{x}}{\cot\left(\frac{\pi x}{2}\right)} \]
8. \[ \lim_{x\to 0} \frac{x \tan^{-1} x}{1 – \cos x} \]
9. \[ \lim_{x\to +\infty} \frac{\ln x}{\sqrt[3]{x}} \]
10. \[ \lim_{x\to 0} \frac{\ln \operatorname{sen} \pi x}{\ln \operatorname{sen} x} \]
11. \[ \lim_{x\to 0} \frac{10^x – 5^x}{x^2} \]
12. \[ \lim_{x\to +\infty} \frac{\ln \ln x}{\sqrt{x}} \]
13. \[ \lim_{x\to \pi} \frac{(x-\pi)^2}{\operatorname{sen}^2 x} \]
14. \[ \lim_{x\to 0} \frac{\tan x – \operatorname{sen} x}{\operatorname{sen}^3 x} \]
15. \[ \lim_{x\to 0} \frac{e^x + e^{-x} – x^2 – 2}{\operatorname{sen}^2 x – x^2} \]
16. \[ \lim_{x\to 0} \frac{x^2 + 2\cos x – 2}{x^4} \]
17. \[ \lim_{x\to \frac{\pi}{4}} \frac{\sec^2 x – 2\tan x}{1 + \cos 4x} \]
18. \[ \lim_{x\to 1} \left[ \frac{1}{\ln x} – \frac{x}{\ln x} \right] \]
19. \[ \lim_{x\to 1} \left[ \frac{x}{x-1} – \frac{1}{\ln x} \right] \]
20. \[ \lim_{x\to 0} \left[ \frac{1}{\operatorname{sen}^2 x} – \frac{1}{x^2} \right] \]
21. \[ \lim_{x\to 0} \left[ \frac{1}{x \operatorname{sen} x} – \frac{1}{x^2} \right] \]
22. \[ \lim_{x\to 0^+} (1 – \cos x) \cot x \]
23. \[ \lim_{x\to \frac{\pi}{4}} (1 – \tan x) \sec 2x \]
24. \[ \lim_{x\to 1} (1 – x) \tan \frac{\pi x}{2} \]
25. \[ \lim_{x\to a} (x^2 – a^2) \tan \frac{\pi x}{2a} \]
26. \[ \lim_{x\to +\infty} x^{\frac{1}{x}} \]
27. \[ \lim_{x\to 0^+} x^{\operatorname{sen} x} \]
28. \[ \lim_{x\to 1} x^{\frac{1}{1-x}} \]
29. \[ \lim_{x\to 0^+} (1 – 2x)^{\frac{1}{x}} \]
30. \[ \lim_{x\to 0^+} (1 + x^2)^{\frac{1}{x}} \]
31. \[ \lim_{x\to 0^+} (\operatorname{sen} x)^{\operatorname{sen} x} \]
32. \[ \lim_{x\to 0} (\operatorname{sen} x)^{x^2} \]
33. \[ \lim_{x\to 0^+} (\operatorname{sen} x)^{\tan x} \]
34. \[ \lim_{x\to 0^+} (\cot x)^{\frac{1}{\ln x}} \]
35. \[ \lim_{x\to 0} \frac{\cosh x – 1}{1 – \cos x} \]
36. \[ \lim_{x\to 0} \frac{\tan^{-1} 2x}{\tan^{-1} 3x} \]
37. \( \lim_{x\to +\infty} (x – \ln(x^2 – 1)) \)

    Sugerencia: \( \ln e^x = x \)
38. \[ \lim_{x\to 0} (1 + \operatorname{senh} x)^{\frac{2}{x}} \]
39. \[ \lim_{x\to 0^+} \left( \frac{1}{x} – \frac{1}{e^x – 1} \right) \]
40. \[ \lim_{x\to +\infty} (e^x – x)^{\frac{1}{x}} \]
41. \( \lim_{x\to +\infty} \frac{(\ln x)^n}{x} \)

    Sugerencia: \( z = \ln x \)
42. \[ \lim_{x\to 0} \frac{\tan^{-1} 3x – 3\tan^{-1} x}{x^3} \]
43. \( \lim_{x\to +\infty} \frac{\ln x}{\sqrt{x}} \)
    Sugerencia: \( z = \sqrt{x} \)

44. Si \( f’ \) es continua, probar: \[ \lim_{h\to 0} \frac{f(x+h) – f(x-h)}{2h} = f'(x) \]

    Sugerencia: Usar regla de L’Hôpital derivando respecto a \( h \).
45. Si \( f» \) es continua, probar: \[ \lim_{h\to 0} \frac{f(x+h) – 2f(x) + f(x-h)}{h^2} = f»(x) \]

    Sugerencia: Usar regla de L’Hôpital derivando 2 veces respecto a \( h \).