Let m and n be two positive integers greater than 1. If lim 0 ( e cos ( n ) - e m ) = - ( e 2 ) , then the value of m n is [2015]

Question

Let $m$ and $n$ be two positive integers greater than 1. If $\lim_{α \to 0} (\frac{e^{\cos (α^{n})} - e}{α^{m}}) = - (\frac{e}{2}),$ then the value of $\frac{m}{n}$ is [2015]

Smart Achievers · Accepted Answer

(2)

$\lim_{α \to 0} \frac{e^{\cos α^{n}} - e}{α^{m}} = - \frac{e}{2}$

$\Rightarrow \lim_{α \to 0} \frac{e [e^{\cos α^{n} - 1} - 1]}{\cos α^{n} - 1} \times \frac{\cos α^{n} - 1}{α^{m}} = - \frac{e}{2}$

$\Rightarrow e \lim_{α \to 0} \frac{- 2 \sin^{2} \frac{α^{n}}{2}}{{(\frac{α^{n}}{2})}^{2}} \times \frac{{(\frac{α^{n}}{2})}^{2}}{α^{m}} = - \frac{e}{2}$

$\Rightarrow \frac{- e}{2} α^{2 n - m} = \frac{- e}{2}$

$\Rightarrow \frac{m}{n} = 2$

Solution

Q.
Let $m$ and $n$ be two positive integers greater than 1. If $\lim_{α \to 0} (\frac{e^{\cos (α^{n})} - e}{α^{m}}) = - (\frac{e}{2}),$ then the value of $\frac{m}{n}$ is [2015]

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Solution

Q. Let m and n be two positive integers greater than 1. If limα→0(ecos(αn)-eαm)=-(e2), then the value of mn is [2015]

Ans. (2) limα→0ecosαn-eαm=-e2 ⇒limα→0e[ecosαn-1-1]cosαn-1×cosαn-1αm=-e2 ⇒elimα→0-2sin2αn2(αn2)2×(αn2)2αm=-e2 ⇒-e2α2n-m=-e2 ⇒mn=2

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Q.
Let $m$ and $n$ be two positive integers greater than 1. If $\lim_{α \to 0} (\frac{e^{\cos (α^{n})} - e}{α^{m}}) = - (\frac{e}{2}),$ then the value of $\frac{m}{n}$ is [2015]