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# 4.8E: Ejercicios para la Sección 4.8

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En los ejercicios 1 - 6, evaluar el límite.

1) Evaluar el límite$$\displaystyle \lim_{x→∞}\frac{e^x}{x}$$.

2) Evaluar el límite$$\displaystyle \lim_{x→∞}\frac{e^x}{x^k}$$.

Responder
$$\displaystyle \lim_{x→∞}\frac{e^x}{x^k} \quad = \quad ∞$$

3) Evaluar el límite$$\displaystyle \lim_{x→∞}\frac{\ln x}{x^k}$$.

4) Evaluar el límite$$\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2}$$.

Responder
$$\displaystyle \lim_{x→a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}$$

5. Evaluar el límite$$\displaystyle \lim_{x→a}\frac{x−a}{x^3−a^3}$$.

6. Evaluar el límite$$\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n}$$.

Responder
$$\displaystyle \lim_{x→a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}$$

En los ejercicios 7 - 11, determina si puedes aplicar directamente la regla de L'Hôpital. Explique por qué o por qué no. Entonces, indica si hay alguna manera de que puedas alterar el límite para que puedas aplicar la regla de L'Hôpital.

7)$$\displaystyle \lim_{x→0^+}x^2\ln x$$

8)$$\displaystyle \lim_{x→∞}x^{1/x}$$

Responder
No se puede aplicar directamente; usar logaritmos

9)$$\displaystyle \lim_{x→0}x^{2/x}$$

10)$$\displaystyle \lim_{x→0}\frac{x^2}{1/x}$$

Responder
No se puede aplicar directamente; reescribir como$$\displaystyle \lim_{x→0}x^3$$

11)$$\displaystyle \lim_{x→∞}\frac{e^x}{x}$$

En los ejercicios 12 - 40, evaluar los límites ya sea con la regla de L'Hôpital o con métodos previamente aprendidos.

12)$$\displaystyle \lim_{x→3}\frac{x^2−9}{x−3}$$

Responder
$$\displaystyle \lim_{x→3}\frac{x^2−9}{x−3} \quad = \quad 6$$

13)$$\displaystyle \lim_{x→3}\frac{x^2−9}{x+3}$$

14)$$\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x}$$

Responder
$$\displaystyle \lim_{x→0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2$$

15)$$\displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}$$

16)$$\displaystyle \lim_{x→π}\frac{x−π}{\sin x}$$

Responder
$$\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1$$

17)$$\displaystyle \lim_{x→1}\frac{x−1}{\sin x}$$

18)$$\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}$$

Responder
$$\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n$$

19)$$\displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}$$

20)$$\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}$$

Responder
$$\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}$$

21)$$\displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}$$

22)$$\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}$$

Responder
$$\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}$$

23)$$\displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}$$

24)$$\displaystyle \lim_{x→1}\frac{x-1}{\ln x}$$

Responder
$$\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1$$

25)$$\displaystyle \lim_{x→0}\,(x+1)^{1/x}$$

26)$$\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}$$

Responder
$$\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}$$

27)$$\displaystyle \lim_{x→0^+}x^{2x}$$

28)$$\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)$$

Responder
$$\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1$$

29)$$\displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}$$

30)$$\displaystyle \lim_{x→0^+}x\ln\left(x^4\right)$$

Responder
$$\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0$$

31)$$\displaystyle \lim_{x→∞}(x−e^x)$$

32)$$\displaystyle \lim_{x→∞}x^2e^{−x}$$

Responder
$$\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0$$

33)$$\displaystyle \lim_{x→0}\frac{3^x−2^x}{x}$$

34)$$\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}$$

Responder
$$\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1$$

35)$$\displaystyle \lim_{x→π/4}(1−\tan x)\cot x$$

36)$$\displaystyle \lim_{x→∞}xe^{1/x}$$

Responder
$$\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞$$

37)$$\displaystyle \lim_{x→0}x^{1/\cos x}$$

38)$$\displaystyle \lim_{x→0^{+} }x^{1/x}$$

Responder
$$\displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0$$

39)$$\displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x$$

40)$$\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x$$

Responder
$$\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}$$

Para los ejercicios 41 - 50, usa una calculadora para graficar la función y estimar el valor del límite, luego usa la regla de L'Hôpital para encontrar el límite directamente.

41) [T]$$\displaystyle \lim_{x→0}\frac{e^x−1}{x}$$

42) [T]$$\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)$$

Responder
$$\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0$$

43) [T]$$\displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}$$

44) [T]$$\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}$$

Responder
$$\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1$$

45) [T]$$\displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}$$

46) [T]$$\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}$$

Responder
$$\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0$$

47) [T]$$\displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)$$

48) [T]$$\displaystyle \lim_{x→0^+}\tan\left(x^x\right)$$

Responder
$$\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1$$

49) [T]$$\displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}$$

50) [T]$$\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}$$

Responder
$$\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2$$

4.8E: Ejercicios para la Sección 4.8 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.