First slide

Introduction to limits

Question

The value of $lim_{x \to 0} {\{\frac{2}{x^{3}} (\tan x - \sin x)\}}^{\frac{2}{x^{2}}}$ is

Moderate

A

e²
B

$\sqrt{e}$
C

$\frac{1}{e}$
D

-1

Solution

$lim_{x \to 0} {\{1 + \frac{2}{x^{3}} (\tan x - \sin x) - 1\}}^{2 / x^{2}}$
$= e^{\lim_{x \to 0}} \frac{[2 (\tan x - \sin x) - x^{3}] \cdot 2}{x^{5}}$
$= e^{\lim_{x \to 0} \frac{2 [2 \{(x - \frac{x^{3}}{3} + \frac{2 x^{5}}{15} - \dots) - (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} \dots - x^{3})\}]}{x^{5}}}$
$= e^{lim_{x \to 0} \frac{2 [2 (\frac{x^{5}}{8} + x^{7} and higher powers of x)]}{x^{5}}}$
$= e^{1 / 2} = \sqrt{e}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App