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  4. displaystyle lim x arrow-1 (1/√|x|- -x ) (where x denotes the fractional part of x ) is equal to

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Q. $\displaystyle\lim _{x \rightarrow-1} \frac{1}{\sqrt{|x|-\{-x\}}}$ (where $\{x\}$ denotes the fractional part of $x$ ) is equal to

Limits and Derivatives

Solution:

L.H.L. $=\displaystyle\lim _{x \rightarrow-1^{-}} \frac{1}{\sqrt{|x|-\{-x\}}}$
$=\displaystyle\lim _{x \rightarrow-1^{-}} \frac{1}{\sqrt{-x-(x+2)}}$
$=\displaystyle\lim _{x \rightarrow-1^{-}} \frac{1}{\sqrt{-2 x-2}}=\infty$
R.H.L. $=\displaystyle\lim _{x \rightarrow-1^{+}} \frac{1}{\sqrt{|x|-\{-x\}}}$
$=\displaystyle\lim _{x \rightarrow-1^{-}} \frac{1}{\sqrt{-x-(x+1)}}$
$=\displaystyle\lim _{x \rightarrow-1^{-}} \frac{1}{\sqrt{-2 x-1}}=1$
Hence, the limit does not exist.