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$\lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{\frac{1}{x^{2}}}=$
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The correct answer is:
$e^{2}$
$\lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{\frac{1}{x^{2}}}$
$=\lim _{x \rightarrow 0} \frac{\left(1+5 x^{2}\right)^{\frac{1}{5 x^{2}} \cdot 5}}{\left(1+3 x^{2}\right)^{\frac{1}{3 x^{2}}} \cdot 3}=\frac{e^{5}}{e^{3}}=e^{5-3}=e^{2}$
$=\lim _{x \rightarrow 0} \frac{\left(1+5 x^{2}\right)^{\frac{1}{5 x^{2}} \cdot 5}}{\left(1+3 x^{2}\right)^{\frac{1}{3 x^{2}}} \cdot 3}=\frac{e^{5}}{e^{3}}=e^{5-3}=e^{2}$
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