Limited yet powerful (VIII)

Calculus Level 4

$\large \lim_{x\to 0} (e^{x^2 /2} \cos x)^{1/x^4}$

If the value of the limit above is equal to $e^{-a/b}$ , where $a$ and $b$ are coprime positive integers, find the value of $a+b$ .

The answer is 13.

2 solutions

Rishi Sharma
Mar 6, 2016

$\lim _{ x->0 }{ { \left( { e }^{ { \frac { x }{ 2 } }^{ 2 } }cosx \right) }^{ \frac { 1 }{ { x }^{ 4 } } } } ={ e }^{ \lim _{ x->0 }{ { \frac { { e }^{ \frac { { x }^{ 2 } }{ 2 } }cosx-1 }{ { x }^{ 4 } } } } }\\ now\quad using\\ { e }^{ x }=\sum _{ n=0 }^{ \infty }{ { \frac { x }{ n! } }^{ n } } \quad and\quad cosx=\sum _{ n=0 }^{ \infty }{ { { \left( -1 \right) }^{ n }\frac { { x }^{ 2n } }{ (2n)! } } } \\ the\quad limit\quad transforms\quad to\quad \\ { e }^{ \lim _{ x->0 }{ \frac { \frac { { -x }^{ 4 } }{ 12 } +.... }{ { x }^{ 4 } } } }={ e }^{ \frac { -1 }{ 12 } }\quad so\quad a=1\quad and\quad b=12\\ hence\quad a+b=13$

Noel Lo
Dec 27, 2015

See below for the step-by-step solution:

Done it same way and used series of tanx at step 7..

rishabh singhal - 4 years, 10 months ago

Limited yet powerful (VIII)

The answer is 13.

2 solutions

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