Evaluate the following limit: _ x 0 a^x+a^ -x -2 x^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{a}^{-\text{x}}-2}{\text{x}^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{a}^{-\text{x}}-2}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{2x}-\text{2a}^\text{x}-2}{\text{a}^2.\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{\text{a}^\text{x}-1}{\text{x}}\Big)^2\times\frac{1}{\text{a}^\text{x}}$
$=(\text{log}_e\text{a})^2\times\frac{1}{\text{a}^0}$
$=(\text{log}_e\text{a})^2$

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