MCQ
The integer $n$ for which $\mathop {\lim }\limits_{x \to 0} \,\frac{{(\cos x - 1)\,(\cos x - {e^x})}}{{{x^n}}}$ is a finite non-zero number is
- A$1$
- B$2$
- ✓$3$
- D$4$
Limit $ = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}\frac{x}{2}}}{{{2^2}{{(x/2)}^2}}}\frac{{{e^x} - \cos x}}{{{x^{n - 2}}}} = \frac{1}{2}\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - \cos x}}{{{x^{n - 2}}}}$
$(n \ne 1$ for then the limit $ = 0)$
$ = \frac{1}{2}\mathop {\lim }\limits_{x \to 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.
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