$\lim\limits_{x\to -\infty}x^4 \arctan(x)$ problems! (easy I guess)

Question

I'm having some trouble calculating this limit:

$$\lim_{x\to -\infty}x^4 \arctan(x)$$

Because I get: $\infty \times (-\frac\pi2) = ?$

Now, I've learned you can't just treat infinity like a number, it's more like an idea or a concept, so you are not supposed to "multiply" $\infty$ with $-\frac\pi2$ and get $-\infty$.

But the answer to this is $-\infty$, so how do I show that in a valid manner?

Answer 1

$$\lim_{x\to -\infty}x^4\arctan(x)=-\infty$$

because $$\lim_{x\to-\infty}x^4=+\infty$$ and

$$\lim_{x\to-\infty}\arctan(x)=\frac{-\pi}{2}<0$$