Need help with limit $\lim_{x \to \infty}x^2 + x\sin (x^2)) = \infty$

Question

I just wanted to make sure I was doing my limits correctly using the formal definition

$$\forall H > 0 \; \exists N > 0 \; s.t. \; \forall x > N \implies f(x) > H$$

Here is what I have done.

Choose $N > \min\{H, 2\}$. Then

$$x > N \implies x \leq x(x - 2) < x(x + \sin(x^2))$$

I'm not sure where to go from here, or even if this is correct at all. I would really appreciate any help. Thanks!

Answer 1

It should be "Choose $N > \max\{H,2\}$" instead [$\max$ instead of $\min$]:

$$x> N \implies x^2+ x \sin x^2 \ge x^2 - x \text{ [make sure you see why]}$$ $$ \ge x(x-1) \ge N(N-1) \ge H(N-1) \ge H(2-1) \ge H.$$

Note that the inequality $N(N-1) \ge H(N-1)$ follows because of the inequality $N>H$, and $H(N-1) \ge H$ because of the inequality $N \ge 2$.