If $\lim_{x \rightarrow \infty} f(x) + f'(x) = 0$, prove $\lim_{x\to \infty} f(x)$ and $\lim_{x \rightarrow \infty} f'(x)$ exist and both equal $0$.

Question

Suppose $\lim_{x \rightarrow \infty} f(x) + f'(x) = 0$.

If we assume $\lim_{x \rightarrow \infty} f(x)$ exists, it's not hard to show that the limit must be 0. But how do we show the limit exists?

Answer 1

In general, let $F(t)=f(t)+f^\prime(t)$. Suppose $\lim_{t\to\infty}F(t)=L$. Let $a=f(0)$.

Then, we have $F(t)=f(t)+f^\prime(t)$ with $f(0)=a$. Thus, by factor integrant, $f(t)=e^{-t}a+e^{-t}\int_0^t e^sF(s)d$.

Thus $f^\prime(t)=-ae^{-t}+F(t)-e^{-t}\int_0^te^{s}F(s)ds$.

If we take $t\to\infty$:

$$\lim_{t\to\infty}f^\prime(t)=0+L-\lim_{t\to\infty}\frac{\int_0^te^{s}F(s)ds}{e^t}=L-\lim_{t\to 0}\frac{e^{t}F(t)}{e^t}=L-L=0.$$

Finally, $\lim_{t\to\infty}f(t)=\lim_{t\to\infty}F(t)-f^\prime(t)=L-0=L$.