Question

Evaluate the following :

${\displaystyle \underset{x\to 0}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]}$

Answer 1

We know that |x| = x if x > 0

= – x if x < 0

∴ ${\displaystyle \underset{x\to 0^{+}}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{x}{x+x^2}}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{x}{x(1+x)}}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{1}{1+x}}$   ...[∵ x → 0, ∴ x ≠ 0]

= ${\displaystyle \frac{\underset{x\to 0}{lim}1}{\underset{x\to 0}{lim}(1+x)}}$

= ${\displaystyle \frac{1}{1+0}}$

= 1

${\displaystyle \underset{x\to 0^{-}}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{x}{-x+x^2}}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{x}{x(-1+x)}}$

= ${\displaystyle \underset{x\to 0}{lim}\frac{1}{-1+x}}$   ...[∵ x → 0, ∴ x ≠ 0]

= ${\displaystyle \frac{\underset{x\to 0}{lim}1}{\underset{x\to 0}{lim}(-1+x)}}$

= ${\displaystyle \frac{1}{-1+0}}$

= – 1

∴ ${\displaystyle \underset{x\to 0^{+}}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]\ne \underset{x\to 0^{-}}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]}$

∴ ${\displaystyle \underset{x\to 0}{lim}\left[\frac{x}{\left|x\right|+x^2}\right]}$ does not exist.