L’Hopital y formas indeterminadas

Ejercicio 6.1

Calcule cada uno de los límites siguientes e indique la forma indeterminada que se presenta.

1

lim_{x \to 1} \frac{1 - x + ln x}{x^{3} - 3 x + 2}

\frac{- 1}{6}

2

lim_{x \to 0} \frac{2 sen (\frac{π}{4} - x) - \sqrt{2}}{x}

- \sqrt{2}

3

lim_{v \to π ∕ 2} sen (cos v) sec v

1

4

lim_{r \to 0^{+}} (e^{r} - 1) ln r

0

5

lim_{r \to π ∕ 2} r tan r - \frac{π}{2} sec r

- 1

6

lim_{q \to 0^{+}} q^{(\frac{3}{4 + ln q})}

e^{3}

7

lim_{t \to {(π ∕ 2)}^{-}} {(tan t)}^{2 t - π}

1

8

lim_{𝜃 \to {(π ∕ 2)}^{-}} {(sen 𝜃)}^{sec 𝜃}

1

9

lim_{n \to \infty} {(1 + \frac{r}{n})}^{nt}

con

r

t

constantes,

t > 0

e^{rt}

10

lim_{x \to 0} \frac{x e^{x} - x}{{sen}^{2} (2 x)}

\frac{1}{4}

11

lim_{x \to 1} (\frac{1}{ln x} - \frac{1}{x - 1})

\frac{1}{2}

12

lim_{z \to + \infty} (z e^{1 ∕ z} - z)

1

13

lim_{x \to {(π ∕ 2)}^{+}} {(x - π ∕ 2)}^{cos x}

1

14

lim_{x \to + \infty} {(\frac{x}{x + 1})}^{x}

e^{- 1}

15

lim_{x \to 0} \frac{ln (1 + x^{2})}{cos (3 x) - e^{- x}}

0

16

lim_{x \to 1} \frac{sen (x - 1)}{ln (2 x - 1)}

\frac{1}{2}

17

lim_{x \to 0} (arcsen x) (csc x)

1

18

lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{x^{2} sec x})

\frac{1}{2}

19

lim_{x \to 0^{+}} x ln (sen x)

0

20

lim_{x \to 0} {(cos (2 x))}^{1 ∕ x^{2}}

e^{- 2}

21

lim_{x \to + \infty} {(1 + \frac{1}{2 x})}^{x^{2}}

+ \infty

22

lim_{x \to 0} \frac{x - tan x}{x - sen x}

- 2

23

lim_{x \to 0} \frac{sen (kx) + tan (nx)}{arctan (nx)}, n \neq 0

\frac{k + n}{n}

24

lim_{y \to 0} {(e^{2 y} - y)}^{y^{- 1}}

e

25

lim_{x \to + \infty} \frac{- 1}{x^{2} {sen}^{2} (\frac{2}{x})}

\frac{- 1}{4}

26

lim_{x \to + \infty} (x - 1) sen (\frac{πx}{x - 1})

- π

27

lim_{x \to 0^{+}} \frac{e^{x} - (1 + x)}{x^{n}}, n = 1, 2, 3, . . .

a): Si n = 1 entonces $lim_{x \to 0^{+}} \frac{e^{x} - (1 + x)}{x^{n}} = 0$ .
b): Si n = 2 entonces $lim_{x \to 0^{+}} \frac{e^{x} - (1 + x)}{x^{n}} = \frac{1}{2}$ .
c): Si n $\geq 3$ entonces $lim_{x \to 0^{+}} \frac{e^{x} - (1 + x)}{x^{n}} = + \infty$ .