Question

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

• A. $3$
• B. $1$
• C. $4$
• D. $2$

Answer 1

The correct option is A $3$

Using expansion of $\cos x$ and $e^x$, we get
$\underset{x\to 0}{lim}\frac{([1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots ]-1)([1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots ]-[1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots ])}{x^n}$
$=\underset{x\to 0}{lim}\frac{(\frac{-x^2}{2!}+\frac{x^4}{4!}-\cdots )(-x-x^2-\cdots )}{x^n}$
$=\underset{x\to 0}{lim}\frac{x^3(\frac{1}{2}+\frac{1}{2}x+\cdots )}{x^n}$
is finite non-zero when $n=3$