Question

The integers n for which $\underset{x\to \infty }{lim}\frac{(cosx-1)(cosx-e^x)}{x^n}$ is a finite non-zero number.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Given,

${lim}_{x\to 0}\frac{(\cos x-1)(\cos x-e^x)}{x^n}$

we know expansion of $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots$

${lim}_{x\to 0}\frac{(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots -1)(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots -e^x)}{x^n}$

${lim}_{x\to 0}\frac{(-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots )(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots -e^x)}{x^n}$

${lim}_{x\to 0}\frac{x^3(-\frac{1}{21}+\frac{x^2}{4!}-\frac{x^3}{6!}+\dots )(\frac{1-e^x}{x^2},-,\frac{1}{2!},+,\frac{x^2}{4!},-,\frac{x^3}{6!},+,,.,)}{x^n}$

for this limit to be finite at $x=0,x^3$ should be eliminated hence $x^n=x^3,n=3$