Lımites trigonometricos

15. Límites trigonométricos

Ejercicio 16.1

Calcule

1: $\lim_{x \to 0} \frac{x^{2} + x sen x}{\cos x - 1}$
$- 4$ .
2: $\lim_{t \to 0} \frac{t - sen (2 t)}{t + sen (3 t)}$
$\frac{- 1}{4}$ .
3: $\lim_{n \to 0} \frac{1 - \cos^{3} n}{{sen}^{2} n}$
$\frac{3}{2}$ .
4: $\lim_{x \to 1} \frac{sen (x - 1)}{\sqrt{x} - 1}$
$2 .$
5: $\lim_{y \to 0} \frac{\tan y - sen y}{{sen}^{3} y}$
$\frac{1}{2}$ .
6: $\lim_{a \to 0} \frac{\sqrt{2} - \sqrt{1 + \cos a}}{{sen}^{2} a}$
$\frac{\sqrt{2}}{8}$ .
7: $\lim_{z \to 0} \frac{\sec (2 z) \cdot \tan (3 z)}{4 z}$
$\frac{3}{4}$ .
8: $\lim_{y \to 0} \frac{\sec y - 1}{y^{3} \csc y}$
$\frac{1}{2}$ .
9: $\lim_{x \to 0} \frac{\tan^{2} x + x^{2} \cos x}{x \tan x}$
$2 .$
10: $\lim_{z \to 0} \frac{\tan (3 z)}{8 z \cos z}$
$\frac{3}{8}$ .
11: $\lim_{x \to 1^{+}} \ln (\frac{4}{sen (3 x - 3)})$
$\infty$ .
12: $\lim_{z \to 0} \frac{sen z - \tan z}{z^{2} \cdot sen (2 z)}$
$\frac{- 1}{4}$ .
13: $\lim_{t \to 0^{-}} \frac{sen (2 t)}{2 \cos (2 t) - 2}$
$\infty$ .
14: $\lim_{x \to {\frac{π}{2}}^{-}} [{(\frac{π}{4})}^{\frac{7}{\tan (x)}} \cdot \frac{sen (2 x - 3 π)}{2 x - π}]$
$1 .$
15: $\lim_{t \to - 4 π ∕ 3} \frac{sen t}{t}$
$\frac{- 3 \sqrt{3}}{8 π} \approx - 0, 206748$ .
16: $\lim_{w \to π ∕ 4} \frac{sen w - \cos w}{1 - \tan w}$
$\frac{- \sqrt{2}}{2}$ .
17: $\lim_{x \to π ∕ 2} \frac{\cot x \cos x}{1 - sen x}$
$2 .$
18: $\lim_{β \to 0} \frac{sen 5 β}{β}$
$5 .$
19: $\lim_{z \to 0} \frac{5 {sen}^{2} (3 z)}{2 z^{2}}$
$\frac{45}{2}$ .
20: $\lim_{u \to 0^{+}} \frac{\sqrt{u}}{sen u}$
$\infty$ .
21: $\lim_{y \to 1} \frac{y^{2} - 1}{sen (𝜋𝑦)}$
$\frac{- 2}{π} \approx - 0, 636620$ .
22: $\lim_{x \to 0} {(\frac{2}{7})}^{\cot | x |}$
$0 .$
23: $\lim_{α \to π} \frac{1 - sen (α ∕ 2)}{π - α}$
$0 .$
24: $\lim_{x \to - π ∕ 6} \frac{\tan (x + π ∕ 6)}{6 x + π}$
$\frac{1}{6}$ .
25: $\lim_{β \to 5 π ∕ 6} 3 sen β - 2 \cos β$
$\frac{3}{2} + \sqrt{3} \approx 3, 23205$ .
26: $\lim_{β \to π ∕ 2} sen (\cos β) \sec^{2} β$
$\infty$ izquierda, $- \infty$ derecha.
27: $\lim_{w \to 0} (\frac{\sec (w) - 1}{w^{2} \csc (w)})$
0.
28: $\lim_{u \to a} \frac{sen u - sen a}{u - a}$ con $a$ constante
$\cos a$ .
29: $\lim_{u \to a} \frac{\cos u - \cos a}{u - a}$ con $a$ constante
$- sen a$ .

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