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