Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$\underset{x\to 0}{lim}\frac{\left(\cos \left(x\right)-1\right)(cosx-e^x)}{x^n}$

$=\underset{x\to 0}{lim}\frac{(1-\frac{x^2}{2!}+\frac{x^4}{4!}+...-1)\left(\right(1-\frac{x^2}{2!}+\frac{x^4}{4!}+...)-(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....\left)\right)}{x^n}$

$=\underset{x\to 0}{lim}\frac{\left(\frac{x^2}{2!}\right)(-\frac{x^2}{2!}+\frac{x^4}{4!}-x-\frac{x^2}{2!}-\frac{x^3}{3!})}{x^n}=\underset{x\to 0}{lim}\frac{(-\frac{x^3}{2!})}{x^n}$

Therefore, $n=3$ for limit to be finite nonzero.
Hence, option 'C' is correct.