Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

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