The value of the limit \lim_{x\to0}(x/\sin x) ^{6\/x^2} is -

KVPY SX SB Exam 2015 Question Paper

Question : 10 of 120

Marks: +1, -0

The value of the limit

\lim↙{x→0}(x/\sin x) ^{6\/x^2}

is -

Solution:

\lim↙{x→0}(x/\sin x) ^{6\/x^2}\;\;\;(1)^∞

\lim↙{e^{x→0}} \; {6/x^2}(x/\sin -1)

\lim↙{e^{x→0}}\; {6/x^2}({x-\sinx} /{\sinx})

\lim↙{e^{x→0}} \; {6/x^2}\{{{x-(x-x^3/3!+x^5/5!...)}/{(x-x^3/3!+x^5/5!...)}}\}

\lim↙{e^{x→0}} \; {6/x^2} { \{ x^3/3!+x^5/5!... \}/{(x-x^3/3!+x^5/5!...)}

\lim↙{e^{x→0}} \; {{6x^3}/x^2} { \{ 1/3!+x^2/5!... \}/{ \{1-x^2/3! ...\}}=e^1

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