# What is the limit of lnx as x approaches 0?

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Steve M Share
Oct 22, 2016

${\lim}_{x \rightarrow 0} \ln x = - \infty$, ie the limit does not exists as it diverges to $- \infty$

#### Explanation:

You may not be familiar with the characteristics of $\ln x$ but you should be familiar with the characteristics of the inverse function, the exponential ${e}^{x}$:

Let $y = \ln x \implies x = {e}^{y}$, so as $x \rightarrow 0 \implies {e}^{y} \rightarrow 0$

You should be aware that ${e}^{y} > 0 \forall y \in \mathbb{R}$,but ${e}^{y} \rightarrow 0$ as $x \rightarrow - \infty$.

The graph of $f \left(x\right) = {e}^{x}$ should help illustrate this:
graph{e^x [-10, 10, -5, 5]}

so if we want ${e}^{y} \rightarrow 0 \implies y \rightarrow - \infty$

Therefore we can conclude that ${\lim}_{x \rightarrow 0} \ln x = - \infty$, ie the limit does not exist as diverges to $- \infty$

The graph of $f \left(x\right) = \ln x$ should help illustrate this:
graph{lnx [-10, 10, -5, 5]}

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