Q112 / 180 · COMEDK 2012 · Mathematics

$\lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{\frac{1}{x^{2}}}=$

  • Correct answer
    A
    $e^{2}$
  • B
    $e$
  • C
    $\frac{1}{e}$
  • D
    $\frac{5}{3}$

Solution

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

Hence, the correct answer is option: 'A'.

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