Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$
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Answer
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{\text{a}^2+\text{x}^2}-\text{a}\big)}{\text{x}^2}\times\frac{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{a}^2+\text{x}^2-\text{a}^2\big)}{\text{x}^2\sqrt{\text{a}^2+\text{x}^2}+\text{a}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^2}{\text{x}^2\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\sqrt{\text{a}^2+\text{x}^2+\text{a}}}$
$=\frac{1}{\text{a}+\text{a}}$
$=\frac{1}{2\text{a}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{\text{a}^2+\text{x}^2}-\text{a}\big)}{\text{x}^2}\times\frac{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{a}^2+\text{x}^2-\text{a}^2\big)}{\text{x}^2\sqrt{\text{a}^2+\text{x}^2}+\text{a}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^2}{\text{x}^2\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\sqrt{\text{a}^2+\text{x}^2+\text{a}}}$
$=\frac{1}{\text{a}+\text{a}}$
$=\frac{1}{2\text{a}}$
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