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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-\text{a}^{\text{-x}}}{\text{x}}=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{2x}-1}{\text{xa}^\text{x}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{2\big(\text{a}^{2\text{x}}-1\big)}{2\text{x}}\lim\limits_{\text{x}\rightarrow0}\frac{1}{\text{a}^\text{x}}$ $=2\text{log}_e 2$

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