Evaluate the following limits: _ x 0 [x |x|+x^2 ]

Question

Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{x}{|x|+x^2}\right]$

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Answer

$ \lim _{x \rightarrow 0}\left[\frac{x}{|x|+x^2}\right]$
$|x|=x ; x \geq 0$
$=-x ; x<0$
$\lim _{x \rightarrow 0^{-}}\left[\frac{x}{|x|+x^2}\right]=\lim _{x \rightarrow 0} \frac{x}{-x+x^2}$
$=\lim _{x \rightarrow 0} \frac{x}{x(-1+x)}$
$=\lim _{x \rightarrow 0} \frac{x}{-1+x} \ldots(\text { As } x \rightarrow 0, x \neq 0)$
$=\frac{1}{-1+0}$
$=-1$
$\lim _{x \rightarrow 0^{+}}\left[\frac{x}{|x|+x^2}\right]=\lim _{x \rightarrow 0} \frac{x}{x+x^2}$
$=\lim _{x \rightarrow 0} \frac{x}{x(1+x)}$
$=\lim _{x \rightarrow 0} \frac{1}{1+x} \ldots(\text { As } x \rightarrow 0, x \neq 0)$
$=\frac{1}{1+0}$
$=1$
$\therefore \quad \lim _{x \rightarrow 0^{-}}\left[\frac{x}{|x|+x^2}\right] \neq \lim _{x \rightarrow 0}\left[\frac{x}{|x|+x^2}\right]$
$\therefore \quad \lim _{x \rightarrow 0} \frac{x}{|x|+x^2} \text { does not exist. }$

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