Prove or find a counter example to the following statements regarding limits

Question

1) If $\lim_{x\to \infty } \frac{f(x)}{x} = \infty$ then $\lim_{x\to \infty } (f(x)-x)= \infty$.

2) If $\lim_{x\to \infty } \frac{f(x)}{x} =1$ then $\lim_{x\to \infty } (f(x)-x)= 0$.

3) If $\lim_{x\to \infty } \frac{f(x)}{x} = 2$ then $\lim_{x\to \infty } (f(x)-x)= \infty$.

I have managed to prove the first statement using $\epsilon-\delta$ formulation and I think that both 2 and 3 are wrong, but can't find a counterexample.

Will you please help me with that?

Thanks !

Answer 1

Take $f(x)=x+\sqrt x$ for the second one. The third one is true.

Answer 2

For $2)$ :

$$\lim_{x \to \infty} \frac{f(x)}{x} = 1 \Rightarrow \lim_{x\to \infty} \Bigg(\frac{f(x)}{x} -1 \Bigg)=0 \Rightarrow\lim_{x \to \infty} \Bigg[\frac{1}{x}\Bigg(f(x)-x\Bigg)\Bigg]=0$$

This does not necessarily means that $\lim_{x\to \infty} [f(x)-x] = 0$, thus it's false.

Regarding $3)$, it's true, can you figure it out yourself ?