Question

Let $f\left(x\right)=(1+\sin x{)}^{\csc x}$, the value of $f\left(0\right)$ so that $f$ is a continuous function is

• A. $e$
• B. $e^{1/2}$
• C. $e^2$
• D. $1$

Answer 1

The correct option is A $e$

$\begin{array}{c}Ifm\left(x\right)andg\left(x\right)aretwofunctionssuchthat\\ {lim}_{x\to a}m\left(x\right)=0={lim}_{x\to a}g\left(x\right)\\ now,\\ {lim}_{x\to a}\left(1+m{\left(x\right)}^{\frac{1}{g\left(x\right)}}\right)={lim}_{x\to a}\frac{m\left(x\right)}{g\left(x\right)}\\ \to itisusefulfro\left(1^{\infty }\right)from\\ here,m\left(x\right)=\sin xg\left(x\right)=\sin x\\ f\left(0\right)=={lim}_{x\to a}{\left(1+\sin x\right)}^{1/\sin x}\\ =e={lim}_{x\to a}\frac{m\left(x\right)}{g\left(x\right)}\frac{\sin x}{\sin x}\\ =e^1=e\end{array}$