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Q. Evaluate $ \underset{x\to 3}{\mathop{\lim }}\,\frac{3-x}{\sqrt{4+x}-\sqrt{1+2x}} $

J & K CETJ & K CET 2014Limits and Derivatives

Solution:

$ \underset{x\to 3}{\mathop{\lim }}\,\,\frac{3-x}{\sqrt{4+x}-\sqrt{1+2x}} $
$=\underset{x\to 3}{\mathop{\lim }}\,\,\frac{(3-x)\,(\sqrt{4+x}+\sqrt{1+2x)}}{{{(\sqrt{4+x})}^{2}}-{{(\sqrt{1+2x})}^{2}}} $
$=\underset{x\to 3}{\mathop{\lim }}\,\frac{(3-x)\,(\sqrt{4+x}+\sqrt{1+2x})}{4+x-1-2x} $
$=\underset{x\to 3}{\mathop{\lim }}\,\frac{(3-x)\,(\sqrt{4+x}+\sqrt{1+2x})}{3-x} $
$=\underset{x\to 3}{\mathop{\lim }}\,\,\sqrt{4+x}+\sqrt{1+2x} $
$=\sqrt{4+3}+\sqrt{1+6} $
$=\sqrt{7}+\sqrt{7}=2\sqrt{7} $