_ x [ 1 + 1 mx ]^x equal to - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to \infty } {\left[ {1 + \frac{1}{{mx}}} \right]^x}$ equal to

✓
${e^{1/m}}$
B
${e^{ - 1/m}}$
C
${e^m}$
D
${m^e}$

✓

Answer

Correct option: A.

${e^{1/m}}$

a
(a) Let $y = \mathop {\lim }\limits_{x \to \,\infty } \,{\left( {1 + \frac{1}{{mx}}} \right)^x} = \mathop {\lim }\limits_{x \to \,\infty } \,{\left( {1 + \frac{1}{{mx}}} \right)^{mx \cdot \frac{1}{m}}}$

$ \Rightarrow \,\,y = {e^{1/m}},\,\,\,\left( {\because \mathop {\lim }\limits_{x \to \,\infty } \,{{\left( {1 + \frac{1}{x}} \right)}^x} = e} \right)$

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