Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Answer 2

$n$ cannot be negative integer for then the limit $= 0$

Limit = $\lim_{x \rightarrow 0}\frac{2\sin^{2}\frac{x}{2}}{2^{2}(x/2)^{2}}\frac{e^{x} - \cos x}{x^{n - 2}} = \frac{1}{2}\lim_{x \rightarrow 0}\frac{e^{x} - \cos x}{x^{n - 2}}$

($n \neq 1$ for then the limit = 0) $= \frac { 1 } { 2 }$ $\lim_{x \rightarrow 0}\frac{e^{x} + \sin x}{(n - 2)x^{n - 3}}.$

So, if $n = 3$, the limit is $\frac{1}{2(n - 2)}$which is finite. If $n = 4,$ the limit is infinite.