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JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Limits XII

Let

m

and

n

be two positive integers greater than 1. If

α \to 0 lim (\frac{e ^{c o s (α^{n})} - e}{α ^{m}}) = - (\frac{e}{2})

then the value of

\frac{m}{n}

is

Answer: 2

Solution

Rewrite the limit as

lim_{α \to 0} \frac{e ( e ^{c o s (α^{n}) - 1} - 1 )}{α ^{m}}

. Using

e^{x} - 1 \sim x

and

cos (α^{n}) - 1 \sim - \frac{( α ^{n} ) ^{2}}{2}

as

α \to 0

, we get

e \cdot 1 \cdot (- \frac{1}{2}) \cdot lim_{α \to 0} \frac{α ^{2 n}}{α ^{m}} = - \frac{e}{2}

. For this limit to equal

- \frac{e}{2}

, we need

lim_{α \to 0} α^{2 n - m} = 1

, which requires

2 n - m = 0

, so

m = 2 n

and

\frac{m}{n} = 2

.

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Integer answer

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