Cal. Diferencial Sec. 4.4

Formas indeterminadas. Regla de L’Hôpital

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

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

    Sugerencia: \(\ln \mathrm{e}^x = x\)

38. \[ \lim\limits_{ x \rightarrow 0} ( 1 + \text{ senh } x )^{2/x} \]
39. \[ \lim\limits_{ x \rightarrow 0^+} \left( \frac{1}{x} - \frac{1}{ \mathrm{e}^x - 1 } \right) \]
40. \[ \lim\limits_{ x \rightarrow +\infty} \left( \mathrm{e}^x - x \right)^{1/x} \]
41. \[ \lim\limits_{ x \rightarrow +\infty} \frac{ (\ln x)^n }{ x } \]

    Sugerencia: \(z = \ln x\)

42. \[ \lim\limits_{ x \rightarrow 0} \frac{ \tan^{-1} 3x - 3\tan^{-1} x }{ x^3 } \]
43. \[ \lim\limits_{ x \rightarrow +\infty} \frac{ \ln x }{ \sqrt[n]{x}} \]

    Sugerencia: \(z = \sqrt[n]{x}\)

44. Si \(f'\) es continua, probar:

    \[ \lim\limits_{ h \rightarrow 0} \frac{ f(x + h) - f(x - h)}{ 2h } = f'(x) \]

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

45. Si \(f''\) es continua, probar:

    \[ \lim\limits_{h \rightarrow 0} \frac{ f(x + h) - 2 f(x) + f(x - h) }{h^2} = f''(x) \]
    \[ \begin{aligned} &\lim\limits_{h \rightarrow 0} \frac{ f(x + h) - 2 f(x) + f(x - h) }{h^2} \\[.5em] &\hspace{10em} = f''(x) \end{aligned} \]

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