Does this limit exist? $\lim_{n\to\infty}\frac{φ(f(n))}{ψ(g(n))}$

Question

EDİTED:

Is this statement true?

Suppose $\left\{f,g,\varphi, \psi \right\}:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ and for $n\to\infty$ $\left\{f(n),g(n),\varphi(n), \psi(n) \right\}\rightarrow \infty$

If $\lim_{n\to\infty}\frac{φ(n)}{ψ(n)}$ doesn't exist, then $\lim_{n\to\infty}\frac{φ(f(n))}{ψ(g(n))}$ doesn't exist too.

I want to know that, if the assumption is wrong, under what conditions is true?

Answer 1

The statement depends on the choice of $f$ and $g$ as mentioned in the comments. So as it stands now, the claim is not true.

Just consider sequences that satisfy $$\frac{\phi(n) } {\psi(n) } = 2+ (-1)^n$$ The limit of that sequence is clearly non-existent. How can we get that? Well just set $\psi(n) =n$ and $$\phi(n) =\psi(n) (2+ (-1)^n) $$ Now set $f(n) =g(n) =2n$ and see that they all satisfy the conditions mentioned by you. Let's see $$\lim_{n\to\infty} \frac{\phi(f(n) ) } {\psi(g(n) ) }=\lim_{n\to\infty} 2+1=3$$ That proves your claim wrong.

Honestly, I can't think of assumptions that make your statement true.