Question:
$\displaystyle\lim_{x\to0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}} = $
$\displaystyle\lim_{x\to0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}} = $
Updated On: May 11, 2024
- $e^{3x}$
- $e^{2}$
- $\frac{1}{e} $
- $\frac{5}{3}$
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The Correct Option is B
Solution and Explanation
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}}}} $
$= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}} = e^{2}$
$\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}}}} $
$= e^{\lim _{x\to 0} \left(\frac{1+5x^{2}}{1+3x^{2}}\right)^{\frac{1}{x^{2}}}} = e^{2}$
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