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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{(\text{a}+\text{x})^2-\text{a}^2}{\text{x}}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{(\text{a}+\text{x}-\text{a})(\text{a}+\text{x}+\text{a})}{\text{x}}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{(\text{x})(2\text{a}+\text{x})}{\text{x}}$

$=\lim\limits_{\text{x}\rightarrow0}(2\text{a}+\text{x})$

$=2\text{a}$

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