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Q. $\displaystyle\lim_{x\to0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}} = $

COMEDKCOMEDK 2012Limits and Derivatives

Solution:

Let $f\left(x\right) = \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}$
$\therefore \:\:\: \lim _{x\to 0} f\left(x\right) =\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}} $
$= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}} $
$= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}} = e^{2}$