Question

If $f\left(x\right)=\{\begin{array}{ccc}(cosx{)}^{1/sinx}& for& x\ne 0\\ k& for& x=0\end{array}$
Then the value of $k$, so that $f$ is continuous at $x=0$ is

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is B $1$

$\underset{x\to 0}{lim}f\left(x\right)=k$

$\underset{x\to 0}{lim}{\left(\cos x\right)}^{\frac{1}{\sin k}}=k$

$1^{\infty }$ form

$e^{\underset{x\to 0}{lim}\frac{\cos x-1}{\sin x}}=e^{\underset{x\to 0}{lim}\frac{-\sin x}{\cos x}}\left(\frac{0}{0}form\right)$

$=e^{-\frac{0}{1}}=1=k$