# Math Help - Natural log limit

1. ## Natural log limit

lim x->0 from the right of ln(4/x^2)

I think it is infinity but I'm not sure how to prove that? I know that if you divide by zero that will give you infinity and I can see that from the graph. However that logarithm being there is making things a bit confusing. I feel like I might be missing a step because of that. Can I just pull out that 4 and take the limit of the rest of the function and then that will give zero?

2. Originally Posted by nautica17
lim x->0 from the right of ln(4/x^2)

I think it is infinity but I'm not sure how to prove that? I know that if you divide by zero that will give you infinity and I can see that from the graph. However that logarithm being there is making things a bit confusing. I feel like I might be missing a step because of that. Can I just pull out that 4 and take the limit of the rest of the function and then that will give zero?
Just pull it apart using the properties of logs.

$ln\left(\frac{4}{x^2}\right)=ln(4)-ln(x^2)$

$=ln(4)-2ln(x)$

So now the limit $2\lim_{x->0^+}ln(x)$ is $-\infty$ right? If you look at a graph of ln(x), you should be able to see that. So the limit is:

$\lim_{n->0^+}ln(4)-2\lim_{n->0^+}ln(x)=\infty$

3. Originally Posted by nautica17
lim x->0 from the right of ln(4/x^2)

I think it is infinity but I'm not sure how to prove that? I know that if you divide by zero that will give you infinity and I can see that from the graph. However that logarithm being there is making things a bit confusing. I feel like I might be missing a step because of that. Can I just pull out that 4 and take the limit of the rest of the function and then that will give zero?
No, The result is not infinity always.
$\frac{a}{0}=\infty$ if $a > 0$.

$\frac{a}{0}=-\infty$ if $a < 0$.

$\frac{a}{0}$ undefined if $a=0$.

What do you mean by "pull out that 4" ??!
In general : $ln (\frac{a}{b}) \neq a ln (\frac{1}{b})$ !!

$ln (\frac{4}{x^2})=ln(4) - 2ln(x)$
Now take the limit, And clearly it infinity.
If you see the graph of $f(x)=ln(\frac{4}{x^2})$, you will see that y-axis is a vertical asymptote.
which prove what do you want.