Solve the following
Question:

$\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{1 / x^{2}}$ is equal to:

  1. (1) $\frac{1}{e}$

  2. (2) $\frac{1}{e^{2}}$

  3. (3) $\mathrm{e}^{2}$

  4. (4) $\mathrm{e}$


Correct Option: , 2

Solution:

Let $\mathrm{R}=\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{3 x^{2}+2}{7 x^{2}+2}-1\right\}}$

$=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{-4 x^{2}}{7 x^{2}+2}\right\}}=e^{\frac{-4}{2}}=e^{-2}=\frac{1}{e^{2}}$

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