# Why is this limit not zero?

1. Jun 19, 2013

### Numnum

1. The problem statement, all variables and given/known data

Why is the limit of $$ln(x)/arctan(x)$$ as x approaches 0 from the right, not zero?

2. Relevant equations

3. The attempt at a solution

I used L'hopital's rule and got zero. But question specifically states that is not the answer.

2. Jun 19, 2013

### jbunniii

$$\lim_{x \rightarrow 0^+} \ln(x) = -\infty$$
whereas
$$\lim_{x \rightarrow 0^+} \arctan(x) = 0$$
so L'Hopital's rule does not apply.

3. Jun 19, 2013

### HallsofIvy

Staff Emeritus
As jbunniii said, L'Hopital's rule does not apply here. Notice that If x= .000001, we have
$$\frac{ln(.000001)}{arctan(.000001)}= \frac{-13.8155}{.000001}= -13815510$$
not anywhere near 0!

4. Jun 19, 2013

### Numnum

Thank you!

But, that's all there is to it? I don't have to simplify anything? Just plug in values?

5. Jun 19, 2013

### HallsofIvy

Staff Emeritus
No, I didn't say that. In order to prove that a limit is a specific number you have to prove that it gets arbitrarily close to that number. My point was that for x very close to 0, the function value is very far away from 0. It is theoretically possible that a function would turn from being around -13 million at x= .000001 to 0 at x= 0, but that would be a very strange function!

6. Jun 19, 2013

### haruspex

It's so obviously unbounded that it's rather a strange question. Are you sure you've stated it correctly?