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Q. The value of $\displaystyle\lim_{x\to2} \frac{5}{\sqrt{2}-\sqrt{x}} $ is

WBJEEWBJEE 2008Limits and Derivatives

Solution:

$\displaystyle\lim_{x\to2} \frac{5}{\sqrt{2}-\sqrt{x}} $

$LHL =\displaystyle \lim_{x\to2^{-}} \frac{5}{\sqrt{2}-\sqrt{x}} $

$ = \displaystyle\lim_{h\to0} \frac{5}{\left(\sqrt{2} -\sqrt{2-h}\right)}\times\frac{\left(\sqrt{2}+\sqrt{2-h}\right)}{\left(\sqrt{2}+\sqrt{2-h}\right)} $

$=\displaystyle \lim_{h\to0} \frac{5\left(\sqrt{2}+\sqrt{2-h}\right)}{2-2-h}=\infty $

$ RHL = \lim_{x\to2^{+}} \frac{5}{\sqrt{2}-\sqrt{x}} $

$=\displaystyle \lim _{h\to 0} \frac{5}{\sqrt{2}-\sqrt{2+h}}\times \frac{\left(\sqrt{2}+\sqrt{2+h}\right)}{\left(\sqrt{2} +\sqrt{2+h}\right)} $

$ =\displaystyle\lim_{h\to 0} \frac{5\left(\sqrt{2}+\sqrt{2+h}\right)}{2-2-h} = -\infty $

$ \because LHL \ne RHL $

$ \therefore $ Limit does not exist