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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