# Question #330d8

Mar 19, 2017

I would say: $- \infty$:

#### Explanation:

Have a look:

Graphically:
graph{(ln(x))/x [-10, 10, -5, 5]}

Mar 19, 2017

Because the function evaluated at 0 results in the indeterminate form $\frac{- \infty}{0}$, one should use L'Hôpital's rule .

#### Explanation:

${\lim}_{x \to 0} \ln \frac{x}{x}$

Apply L'Hôpital's rule:

${\lim}_{x \to 0} \frac{\frac{d \left(\ln \left(x\right)\right)}{\mathrm{dx}}}{\frac{d \left(x\right)}{\mathrm{dx}}} =$

${\lim}_{x \to 0} \frac{\frac{1}{x}}{1} =$

${\lim}_{x \to 0} \frac{1}{x}$

This limit is known to be unbounded

Mar 19, 2017

${\lim}_{x \rightarrow {0}^{+}} \ln \frac{x}{x} = - \infty$

#### Explanation:

${\lim}_{x \rightarrow {0}^{+}} \ln x = - \infty$ and

${\lim}_{x \rightarrow {0}^{+}} \frac{1}{x} = \infty$.

So the product has a limit of $- \infty$