Finding $\lim_{x\to\infty}\frac{g(x)}{x}$

Question

Let $f : [0,\infty ) \to \mathbb{R}$ be continuous and periodic. Let $g(x)=\int_0^xf(t)dt$, for all $x\geq 0.$ Suppose $\lim_{x\to\infty}f(x)=L$. Find $$\lim_{x\to\infty}\frac{g(x)}{x}.$$

I claimed, $f$ is a constant, as it is periodic and has limit at infinity, so $g(x)=Lx$. Hence $$\lim_{x\to\infty}\frac{g(x)}{x}=L$$

Is it a valid argument?

Answer 1

If $f$ is continuous then $g(x)$ is differentiable thus

$$\lim_{x \to + \infty} \frac{g(x)}{x}=L$$ from L'Hospital's rule.