Let m and n be two positive integers greater than 1. If _ α → 0 ( e ^ ( α ^ n ) - e α ^ m ) = - ((e)/(2)) then the value of (m)/(n) is

Correct answer: 0 _ α → 0 e ^ cos α ^ n - e α ^ m = - (e)/(2) ⇒ Using L'Hospital's Rule _ α → 0 e ^ ( α ^ n ) ⋅ - ( α ^ n ) ⋅ n α ^ n - 1 m ⋅ α ^ m - 1 = (- e)/(2) ⇒ (m)/(n) = 2 .

JEE Advanced 2015Paper 2

Exam: JEE Advanced
Year: 2015
Paper: Paper 2
Subject: Mathematics
Chapter: Limits

JEE Advanced 2015 · Paper 2 · Mathematics · Limits

Let m and n be two positive integers greater than 1. If _ α → 0 ( e...

Let m and n be two positive integers greater than 1. If

\lim_{\alpha \to 0}

\left(\dfrac{{e}^{\cos ( {\alpha}^{n} )} - e}{{\alpha}^{m}}\right) = -

\left(\dfrac{e}{2}\right)

then the value of

\dfrac{m}{n}

Correct answer

Step-by-step solution

\lim_{\alpha \to 0} \dfrac{{e}^{cos {\alpha}^{n}} - e}{{\alpha}^{m}} = - \dfrac{e}{2}

⇒

Using L'Hospital's Rule

\lim_{\alpha \to 0} \dfrac{{e}^{\cos \left({\alpha}^{n}\right)} &sdot; - \sin \left({\alpha}^{n}\right) &sdot; n{\alpha}^{n - 1}}{m &sdot; {\alpha}^{m - 1}} = \dfrac{- e}{2}

⇒ \dfrac{m}{n} = 2

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