Evaluate the following limit: _ x 0 5^x+3^x+2^x-3 x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{5^\text{x}+3^\text{x}+2^\text{x}-3}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{5^\text{x}+3^\text{x}+2^\text{x}-3}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{5^\text{x}-1}{\text{x}}+\lim\limits_{\text{x}\rightarrow0}\frac{3^\text{x}-1}{\text{x}}+\lim\limits_{\text{x}\rightarrow0}\frac{2^\text{x}-1}{\text{x}}$
$=\text{log5}+\text{log3}+\text{log2}$
$=\text{log 30}$

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