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$\lim _{x \rightarrow 0} \frac{2 x}{|x|+x^2}=$
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Limit does not exists
$\lim _{x \rightarrow 0} \frac{2 x}{|x|+x^2}$
L.H.L $\lim _{x \rightarrow 0^{-}} \frac{2 x}{-x+x^2}=\lim _{x \rightarrow 0^{-}} \frac{2}{-1+x}=-\infty$
R.H.L $\lim _{x \rightarrow 0^{+}} \frac{2 x}{x+x^2}=\lim _{x \rightarrow 0^{+}} \frac{2}{1+x}=\infty$
$\because$ L.H.L $\neq$ R.H.L $\Rightarrow$ limit does not exist
L.H.L $\lim _{x \rightarrow 0^{-}} \frac{2 x}{-x+x^2}=\lim _{x \rightarrow 0^{-}} \frac{2}{-1+x}=-\infty$
R.H.L $\lim _{x \rightarrow 0^{+}} \frac{2 x}{x+x^2}=\lim _{x \rightarrow 0^{+}} \frac{2}{1+x}=\infty$
$\because$ L.H.L $\neq$ R.H.L $\Rightarrow$ limit does not exist
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