Graph of function

Question

I know this is an elementary question, but I'm having some problems in graphing the following function:

$$f(x) = \arctan(\ln x)$$

I know I have to evaluate these limits:

$$\lim\limits_{x \to 0^+} f(x), \lim\limits_{x \to 0^-} f(x)$$

$$\lim\limits_{x \to \infty} f(x), \lim\limits_{x \to -\infty} f(x)$$

But I can't figure out a way to manipulate the expression in order to get a decent result.

Thanks in advance.

Answer 1

Hint: For something like $\lim_{x\to 0^{+}}\arctan(\ln(x))$, think about what happens on the inside first: $\ln(x)\to-\infty$. So then ask, what is $\lim_{u\to-\infty}\arctan(u)$? For this, you can either think about what $\arctan(x)$ means... or if you just know what the graph looks like (it has two horizontal asymptotes).