A function $f : (a, ∞) → \Bbb R$ is differentiable on $(a, ∞)$, and satisfies $$\lim_{x→∞}(f'(x) + αf(x)) = 0$$ where $α$ is a positive constant
Prove that $\lim_{x→∞}f(x) = 0$.
I think that I should write $f(x)$ as $f(x) = \dfrac{f(x) + \alpha f(x)}{1 +\alpha}$
Is it correct? Any help would be appreciated.
Write $$f(x)=\frac{e^{\alpha x}f(x)}{e^{\alpha x}}$$ By L'Hospital's rule, $$\lim_{x \to \infty} f(x)=\lim_{x \to \infty} \frac{e^{\alpha x} f'(x)+\alpha f(x)e^{\alpha x}}{\alpha e^{\alpha x }}=\frac{1}{\alpha}\lim_{x \to \infty } [f'(x)+\alpha f(x)]=0$$