Question

The integer $n$ for which $\displaystyle\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}$ is a finite non-zero number, is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

$\displaystyle\lim _{x \rightarrow 0} \frac{(\cos x-1)\left(\cos x-e^{x}\right)}{x^{n}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\left(-2 \sin ^{2} \frac{x}{2}\right)\left\{\left(1-\frac{x^{2}}{2}+\frac{x^{4}}{4 !}-\ldots .\right) -\left(1+x+\frac{x^{2}}{2 !}+\frac{z^{3}}{3 !}+\ldots .\right)\right\}}{x^{n}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\left(-2 \sin ^{2} \frac{x}{2}\right)\left(-x-\frac{2 x^{2}}{2 !}-\frac{x^{3}}{3 !}-\ldots\right)}{4\left(\frac{x}{2}\right)^{2} x^{n-2}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\sin ^{2} \frac{x}{2}\left(1+x+\frac{x^{2}}{3 !}+\ldots\right)}{2\left(\frac{x}{2}\right)^{2} x^{n-3}}$
Above limit is finite, if $n-3=0$, i.e. $n=3$.