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Q. $ \displaystyle \lim_{x \to 0} $ $ \frac{x}{\left|x\right|+x^{2}} $ is equal to

UPSEEUPSEE 2012

Solution:

$\displaystyle\lim _{x \rightarrow 0^{-}} f(x)=\displaystyle\lim _{h \rightarrow 0} \frac{0-h}{h+h^{2}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{-1}{1+h}=-1$
and $\displaystyle\lim _{x \rightarrow 0^{+}} f(x)=\displaystyle\lim _{h \rightarrow 0} \frac{h}{h+h^{2}}=\displaystyle\lim _{h \rightarrow 0} \frac{1}{1+h}=1$
Hence, limit does not exist.