$\lim _{x \rightarrow 0} \frac{2 x}{|x|+x^2}=$

$\lim _{x \rightarrow 0} \frac{2 x}{|x|+x^2}=$
  1. Limit exists
  2. Limit does not exists
  3. $2$
  4. $-2$

Solution

$\lim _{x \rightarrow 0} \frac{2 x}{|x|+x^2}$ 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

Asked in: MHT CET 2022 (07 Aug Shift 2)

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