_ m , ( x m )^m = - Vidyadip

Question

$\mathop {\lim }\limits_{m \to \infty } \,{\left( {\cos \frac{x}{m}} \right)^m} = $

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Answer

d
(d) $\mathop {\lim }\limits_{m \to \infty } {\left( {\cos \frac{x}{m}} \right)^m} = \mathop {\lim }\limits_{m \to \infty } {\left[ {1 + \left( {\cos \frac{x}{m} - 1} \right)} \right]^m}$

$ = \mathop {\lim }\limits_{m \to \infty } {\left[ {1 - \left( { - \cos \frac{x}{m} + 1} \right)} \right]^m}$

$ = \mathop {\lim }\limits_{m \to \infty } {\left[ {1 - 2{{\sin }^2}\frac{x}{{2m}}} \right]^m}$

$ = {e^{\mathop {\lim }\limits_{m \to \infty } - \left( {2{{\sin }^2}\frac{x}{{2m}}} \right)\,m}}$

$ = {e^{\mathop {\lim }\limits_{m \to \infty } - 2{{\left( {\frac{{\sin \frac{x}{{2m}}}}{{x/2m}}} \right)}^2}\left( {\frac{{{x^2}}}{{4{m^2}}}} \right)\,m}}$

$ = {e^{ - 2\mathop {\lim }\limits_{m \to \infty } \frac{{{x^2}}}{{4m}}}} = {e^0} = 1$.

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