If $\lim_{x\rightarrow\infty} f(x) $ tends to infinity does it imply that $f(x)$ is eventually monotonically increasing

Question

If $\lim_{x\rightarrow\infty} f(x)=\infty $ exists does it imply that $f(x)$ is eventually monotonically increasing.How to prove if true ?

I want to apply this on polynomials.

Answer 1

No. $x + 10 \sin(x)$ is a counterexample.

Answer 2

The general statement, as given, is not true but it is true for polynomials! If p is a polynomial of degree n, then p' is a polynomial of degree n-1. p' has a finite number, at most n-1, of real zeros so has a largest zero. If z is larger than that, p' does not change sign so p is monotone increasing or decreasing. (However, for polynomials, $\lim_{x\to \infty} f(x)$ never exists.)