limits - Proving $lim_{xtoinfty} (x^2 +1)(frac{pi}{2} - arctan{x})$ doesn't exist.

How can I show that

$$\lim_{x\to\infty} (x^2 +1)(\frac{\pi}{2} - \arctan{x})$$ doesn't exist? I used the fact that $$\arctan{x}\ge x-\frac{x^3} {3},$$ so the initial limit is less than $$\lim_{x\to\infty} \frac{x^5}{3} +O(x^4),$$ therefore the limit tends to infinity.

Is this enough? If not, then how can I show this rigorously?