calculus - If $lim limits_{xto infty} f'(x)=ell$ then find $ limlimits _{x to infty} frac{f(6x)}{x}$.

Let $f:[0,\infty)\to\mathbb{R}$ be continuously differentiable. If $\lim _{x\to \infty} f'(x)=\ell$ for real number $\ell$, then find $$ \lim _{x \to \infty} \frac{f(6x)}{x}$$

Mt attempt:
By L'Hospital rule
$$\lim _{x \to \infty} \frac{f(6x)}{x}=6\lim _{x \to \infty} f'(6x)=6\ell.$$ Am I right?

Answer

Without Hospital Only Using the fundamental theorem of Calculus we have,

$$f(6x) =f(0) +\int_0^x 6f'(6t)dt\overset{t=xu}{=}f(0) +x\int_0^1 6f'(6\color{blue}{xu})dt,~~ x>0.$$

Thus, since $[0,1]$ is compact and $\lim\limits_{x\to\infty}f'(x)= \ell$ we get,
$$\lim _{x \to \infty} \frac{f(6x)}{x} =\lim _{x \to \infty}\left(\frac{f(0)}{x}+ 6\int_0^1 f'(6\color{blue}{xu})du\right) =6\ell$$