A) \[{{e}^{2}}\]
B) \[e\]
C) \[{{e}^{-2}}\]
D) \[{{e}^{-1}}\]
Correct Answer: A
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\,{{\left( \frac{1+5{{x}^{2}}}{1+3{{x}^{2}}} \right)}^{1/{{x}^{2}}}}=\frac{\underset{x\to 0}{\mathop{\lim }}\,\,\,{{\left[ {{(1+5{{x}^{2}})}^{1/5{{x}^{2}}}} \right]}^{5}}}{\underset{x\to 0}{\mathop{\lim }}\,\,\,{{\left[ {{(1+3{{x}^{2}})}^{1/3{{x}^{2}}}} \right]}^{3}}}=\frac{{{e}^{5}}}{{{e}^{3}}}={{e}^{2}}\]. \[[\because \,\,\,\underset{x\to 0}{\mathop{\lim }}\,\,{{(1+x)}^{1/x}}=e]\]
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