Simple question, can't seem to find an intuitive explanation anywhere. But the question is as follows, why does Why is
$$
\lim_{x\to 0^+} \ln x = -\infty.
$$
and Why is
$$
\lim_{x\to \infty} \ln x = \infty.
$$
Honestly the second one is a tad bit easier to understand. You are trying to find the natural log of an infinite number so the number you would be left with, would also be infinite, but the first one ($\ln{0}$) just doest not make sense for me.
Thanks in advance.
Answer
I do not believe a statement like $\ln \infty = \infty$
has an exact, universally accepted mathematical meaning.
What I think we can agree on is that whatever that formula represents,
it is based on the truth of the generally-accepted
true mathematical statement,
$$
\lim_{x\to\infty} \ln x = \infty
$$
which in turn is not really an equation, but is rather a way of
saying that if we want to the value of $\ln x$ to be greater than
some positive number (choose any number you want),
all we have to do is to ensure that $x$ is a large enough positive number.
There is no upper bound on how large we can force $\ln x$ to be,
and all we have to do in order to make $\ln x$ "large enough"
is name a number $N$ and assert that $x > N$.
So what we're really trying to explain is why
$$
\lim_{x\to 0^+} \ln x = -\infty.
$$
That is, to force $\ln x$ to be less than some arbitrarily large negative
number, all we have to do is make $x$ close enough to (but greater than) $0$.
Now, we know that
$$
\ln x = - \ln \frac1x,
$$
and we know that
$$
\lim_{x\to 0^+} \frac 1x = \infty.
$$
So by making $x$ close to zero, but positive,
we can make $\frac1x$ be as large a positive number as we like.
And therefore we can make $\ln \frac1x$ be as large a positive number
as we need to.
But if $\ln \frac1x$ is a large positive number,
then $\ln x = -\ln \frac1x$ is a large negative number.
And that's how we know that
$$
\lim_{x\to 0^+} \ln x = -\infty.
$$
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