The value of $\lim _{x \rightarrow 0} \frac{x}{|x|+x^2}$ is .

The value of $\lim _{x \rightarrow 0} \frac{x}{|x|+x^2}$ is .
  1. 1
  2. -1
  3. 0
  4. does not exist.

Solution

$\begin{aligned} & \text { L.H.L. }=\lim _{x \rightarrow 0^{-}} \frac{x}{|x|+x^2} \\ & =\lim _{x \rightarrow 0^{-}} \frac{x}{-x+x^2} \\ & =\lim _{x \rightarrow 0^{-}} \frac{1}{x-1}=-1 \\ & \text { R.H.L. }=\lim _{x \rightarrow 0^{+}} \frac{x}{|x|+x^2} \\ & =\lim _{x \rightarrow 0^{+}} \frac{x}{x+x^2} \\ & =\lim _{x \rightarrow 0^{+}} \frac{1}{1+x}=1 \\ & =\text { L.H.L. } \neq \text { R.H.L. } \end{aligned}$ $\therefore \quad$ Limit does not exist.

Asked in: MHT CET 2024 (02 May Shift 1)

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