Let $f$ be a real-variable function such that $\lim f(x) = L$ as $x\rightarrow +\infty$ where $L\in\mathbb{R}$. Also, let $c>0$ be a constant. My question is:
Is $\lim f(c x) = L$ as $x\rightarrow +\infty$?
My attempt: Define $c x = t$. Since $x\rightarrow +\infty$ and $c>0$ then $t\rightarrow +\infty$, so $\lim f(t) = L$ as $x\rightarrow +\infty$. Is this it correct?
$$\forall c\in\Bbb R^+; x\to \infty \iff cx\to\infty$$
That's all you need.