limit of continuously differentiable function and its derivative

Question

Let $f$ be a continuously differentiable function on $\mathbb{R}$. Suppose that $L=\lim_{x\to \infty}(f(x)+f'(x))$ exists.

If $ 0\lt L \lt\infty$ ,then which of the following statements is/are correct?

1.If $\lim_{x\to \infty}f'(x)$ exists then it is $0$

2.If $\lim_{x\to \infty}f(x)$ exists then it is $L$

3.If $\lim_{x\to \infty}f'(x)$ exists then $\lim_{x\to \infty} f(x) =0$

4.If $\lim_{x\to \infty}f(x)$ exists then $\lim_{x\to \infty}f'(x) =L$

Let $f(x)=1+ 1/x$ then $\lim_{x\to \infty}f(x)=1$ and $\lim_{x\to \infty}f'(x)=0$ so $L=1$

using this example 1,2 correct but this function is not differentiable at x=0 so how to proceed..

Answer 1

You made a very common, but also very bad, logical error:

You cannot prove a statement by providing an example, you can only disprove it.

---

For example:

My claim:

• If a function is continuous, then it is bounded

My proof:

• Take the example $f(x)=\sin x$. Clearly, $f$ is bounded, so my claim is correct.

Or even worse, my claim:

• All white people live in Europe

My proof:

• Take for example me, a white person. I live in Europe, so my claim is correct.

---

To actually prove the claim, you need to prove it for every function $f$.

To do that, here's a couple of hints:

• If $\lim g(x)$ and $\lim h(x)$ both exist, then $\lim(g(x)-h(x))$ also exists.
• From that, you can conclude that if $\lim g(x)+h(x)$ exists and $\lim g(x)$ exists, then $\lim h(x)$ exists (because $h(x)= (g(x)+h(x)) - g(x)$)
• for $(1)$, think about what happens if the limit of $f'$ is not $0$.
• for $(2)$, you can easily translate this into $(1)$.

Answer 2

If $\lim_{x\rightarrow\infty}f(x)$ exists and equals $l\in\mathbb{R}$ then $$l=\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty}\frac{e^xf(x)}{e^x}=\lim_{x\rightarrow\infty}\frac{\frac{d}{dx}e^xf(x)}{\frac{d}{dx}e^x}=\lim_{x\rightarrow\infty}(f(x)+f'(x))$$ or $l=L$.