Question

The value of $\underset{x\to 0}{lim}(1+x^2+\sin x{)}^{\frac{3}{\tan x}}$ is

• A. $e$
• B. 1
• C. 0
• D. $e^3$

Answer 1

The correct option is D $e^3$

Let $L=\underset{x\to 0}{lim}(1+x^2+\sin x{)}^{\frac{3}{\tan x}}$
Clearly form of the limit is $1^{\infty }$.
$\therefore L=\underset{x\to 0}{lim}(1+x^2+\sin x{)}^{\frac{1}{x^2+\sin x}\cdot \frac{3(x^2+\sin x)}{\tan x}}$
$=e^{3\underset{x\to 0}{lim}\frac{x+\frac{\sin x}{x}}{\frac{\tan x}{x}}}=e^{3\cdot \frac{0+1}{1}}=e^3$