Question

If the function $f\left(x\right)=\{\begin{array}{cc}\frac{1-\sqrt{\cos x}}{x^2},& x\ne 0\\ k,& x=0\end{array}$ is continuous at $x=0,$ then the value of $4k+3$ is

Answer 1

For continuity at $x=0$
$\underset{x\to 0}{lim}\frac{1-\sqrt{\cos x}}{x^2}=f\left(0\right)$

$\Rightarrow \underset{x\to 0}{lim}\frac{1-\sqrt{\cos x}}{x^2}\times \frac{1+\sqrt{\cos x}}{1+\sqrt{\cos x}}=k$
$\Rightarrow \underset{x\to 0}{lim}\frac{1-\cos x}{x^2(1+\sqrt{\cos x})}=k$
$\Rightarrow \underset{x\to 0}{lim}\frac{2{\sin }^2\frac{x}{2}}{2x^2}=k$
$\Rightarrow k=\frac{1}{4}$
$\therefore 4k+3=4$