JEE Advanced 2015 · Paper 2 · Mathematics Q45

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MathematicsInteger+4Question 45 of 60

Let $m$ and $n$ be two positive integers greater than 1. If
$\lim_{\alpha\to0}\left(\frac{e^{\cos(\alpha^n)}-e}{\alpha^m}\right)=-\left(\frac e2\right)$
then the value of $\dfrac mn$ is

Your answer

Answer and worked solution

Official answer: 2

Expand.
$e^{\cos\alpha^n}-e=e\left(e^{\cos\alpha^n-1}-1\right)\approx e(\cos\alpha^n-1)\approx-\frac e2\alpha^{2n}.$

Match. The limit equals $-\frac e2$ only when $m=2n$ , so $\dfrac mn=2$ .

Answer: 2

Topics:Limits

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JEE Advanced 2015 · Paper 2, Mathematics question 45