Evaluate the value of ((x+4)^2-4)/x as x approaches to 0?

May 5, 2018

Does not exist.

Explanation:

${\lim}_{x \rightarrow 0} \frac{{\left(x + 4\right)}^{2} - 4}{x}$ =^((12/0))?

• If $x \to {0}^{+}$ , $x > 0$ then

${\lim}_{x \rightarrow {0}^{+}} \frac{{\left(x + 4\right)}^{2} - 4}{x}$ ${=}^{\left(\frac{12}{0} ^ \left(+\right)\right)}$ $+ \infty$

• If $x \to {0}^{-}$, $x < 0$ then

${\lim}_{x \rightarrow {0}^{-}} \frac{{\left(x + 4\right)}^{2} - 4}{x}$ ${=}^{\left(\frac{12}{0} ^ \left(-\right)\right)}$ $- \infty$

Graphical help

May 5, 2018

$4$

Explanation:

Let,

${\lim}_{x \to 0} f \left(x\right) = {\lim}_{x \to 0} \frac{{\left(x + 4\right)}^{2} - 4}{x}$

If $x \to {0}^{-} , t h e n , \frac{1}{x} \to - \infty \implies {\lim}_{x \to {0}^{-}} f \left(x\right) \to - \infty \mathmr{and}$

If $x \to {0}^{+} , t h e n , \frac{1}{x} \to + \infty \implies {\lim}_{x \to {0}^{+}} f \left(x\right) \to + \infty$

Hence,

${\lim}_{x \to 0} f \left(x\right)$ does not exist.