The integral value of $\mathrm{n}$ for which $\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos…
The integral value of $\mathrm{n}$ for which $\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}$ is a finite non zero real number is
$4$
$3$
$2$
$1$
Solution
$\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}$
$\begin{aligned} & \lim _{x \rightarrow 0} \frac{2 \sin ^2 x / 2\left(e^x-1+1-\cos n\right)}{x^n} \\ & \lim _{x \rightarrow 0} \frac{1}{2}\left[\frac{\sin (x / 2)}{x / 2}\right]^2 \frac{\left(e^x-1+2 \sin ^2 x / 2\right)}{x^{n-2}}\end{aligned}$
It is non-zero if $\mathrm{n}-2=1 \Rightarrow \mathrm{n}=3$