Let m & n be two positive numbers greater than 1 . If lim_(p→0) ((e^(cos (p^n )) −e)/p^m ) = (1/2)e then (n/m)=?


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Question Number 162399 by cortano last updated on 29/Dec/21

$L e t m & n b e t w o p o s i t i v e n u m b e r s$ $g r e a t e r t h a n 1 . I f \lim_{p \to 0} \frac{e^{\cos (p^{n})} - e}{p^{m}} = \frac{1}{2} e$ $t h e n \frac{n}{m} = ?$
Commented by blackmamba last updated on 29/Dec/21

$\lim_{p \to 0} \frac{e^{\cos (p^{n})} - e}{p^{m}} = e \lim_{p \to 0} \frac{e^{\cos (p^{n}) - 1} - 1}{p^{m}} = \frac{1}{2} e$ $\lim_{p \to 0} \frac{e^{\cos (p^{n}) - 1} - 1}{\cos (p^{n}) - 1} \times \lim_{p \to 0} \frac{\cos (p^{n}) - 1}{p^{m}} = \frac{1}{2}$ $[n o t i c e t h a t \lim_{x \to 0} \frac{e^{x} - 1}{x} = 1]$ $[\lim_{p \to 0} \frac{e^{\cos (p^{n}) - 1} - 1}{\cos (p^{n}) - 1} = 1]$ $\Leftrightarrow \lim_{p \to 0} \frac{\cos (p^{n}) - 1}{p^{m}} = \frac{1}{2}$ $\Leftrightarrow \lim_{p \to 0} \frac{- 2 \sin^{2} (\frac{p^{n}}{2})}{p^{m}} = \frac{1}{2}$ $\Leftrightarrow \lim_{p \to 0} \frac{{(\frac{\sin (\frac{p^{n}}{2})}{(\frac{p^{n}}{2})})}^{2} . (\frac{p^{2 n}}{4})}{p^{m}} \overset{?}{=} - \frac{1}{4}$ $2 n = m \Rightarrow \frac{n}{m} = \frac{n}{2 m} = \frac{1}{2}$
	Answered by qaz last updated on 29/Dec/21

	$\lim_{p \to 0} \frac{e^{\cos (p^{n})} - e}{p^{m}}$ $= \lim_{p \to 0} \frac{e (e^{\cos (p^{n}) - 1} - 1)}{p^{m}}$ $= e \lim_{p \to 0} \frac{\cos (p^{n}) - 1}{p^{m}}$ $= e \lim_{p \to 0} \frac{- \frac{1}{2} p^{2 n}}{p^{m}}$ $= - \frac{1}{2} e$ $\Rightarrow \frac{n}{m} = \frac{1}{2}$
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