Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{9^\text{x}-2.6^\text{x}+4^\text{x}}{\text{x}^2}$
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Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{9^\text{x}-2.6^\text{x}+4^\text{x}}{\text{x}^2}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\big(3^\text{x}\big)^2-2.3^\text{x}2^\text{x}+\big(2^\text{x}\big)^2}{\text{x}^2} $ $=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{3^\text{x}-2^\text{x}}{\text{x}}\Big)^2$ $=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{3^\text{x}-1}{\text{x}}-\lim\limits_{\text{x}\rightarrow0}\Big(\frac{2^\text{x}-1}{\text{x}}\Big)\Big)^2$ $=\Big(\text{log}\frac{3}{2}\Big)^2$
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