Evaluate the following limit: _ x 4 2- x 4-x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow4}\frac{2-\sqrt{\text{x}}}{4-\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow4}\frac{2-\sqrt{\text{x}}}{4-\text{x}}$$\lim\limits_{\text{x}\rightarrow4}\frac{\big(2-\sqrt{\text{x}}\big)\big(2+\sqrt{\text{x}}\big)}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$
$\lim\limits_{\text{x}\rightarrow4}\frac{(2)^2-\big(\sqrt{\text{x}}\big)^2}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$
$\lim\limits_{\text{x}\rightarrow4}\frac{(4-\text{x})}{(4-\text{x})\big(2+\sqrt{\text{x}}\big)}$
$=\frac{1}{2+\sqrt{4}}$
$=\frac{1}{2+2}$
$=\frac14$

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