Cal. Diferencial Sec. 4.4

Formas indeterminadas. Regla de L’Hôpital

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

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

    Sugerencia: $\ln {\mathrm{e}}^x=x$

38. $$\underset{x\to 0}{lim}(1+\text{ senh }x{)}^{2/x}$$
39. $$\underset{x\to 0^{+}}{lim}(\frac{1}{x}-\frac{1}{{\mathrm{e}}^x-1})$$
40. $$\underset{x\to +\mathrm{\infty }}{lim}{({\mathrm{e}}^x-x)}^{1/x}$$
41. $$\underset{x\to +\mathrm{\infty }}{lim}\frac{(\ln x{)}^n}{x}$$

    Sugerencia: $z=\ln x$

42. $$\underset{x\to 0}{lim}\frac{{\tan }^{-1}3x-3{\tan }^{-1}x}{x^3}$$
43. $$\underset{x\to +\mathrm{\infty }}{lim}\frac{\ln x}{\sqrt[n]{x}}$$

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

44. Si $f^{\prime }$ es continua, probar:

    $$\underset{h\to 0}{lim}\frac{f(x+h)-f(x-h)}{2h}=f^{\prime }(x)$$

    Sugerencia: Usar regla de L’Hôpital derivando respecto a $h$.

45. Si $f^{″}$ es continua, probar:

    $$\underset{h\to 0}{lim}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f^{″}(x)$$
    $$\begin{array}{rl}& \underset{h\to 0}{lim}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\\ & =f^{″}(x)\end{array}$$

    Sugerencia: Usar regla de L’Hôpital derivando 2 veces respecto a $h$.