Simplifying $f(n)$ by substituting, for $n$, an appropriately chosen function $n(x)$ to observe limiting behaviour of $f(n)$. Is this justified?

Question

Say, I'm comparing two functions

$f(n) = (ln(n))^2$ and $ g(n) = n^{0.01}$

as $n \rightarrow \infty$, by evaluating

$\lim_{n \rightarrow \infty } \frac{f(n)}{g(n)} = \lim_{n \rightarrow \infty } \frac{ln(n)^2}{n^{0.01}} $.

My question:

Am I justified in simplifying this by expressing $n$ as a function of $x \in \Re$, such that

$n(x) = e^x$,

so that the original limit becomes

$\lim_{x \rightarrow \infty } \frac{f(n(x))}{g(n(x))} = \lim_{x \rightarrow \infty } \frac{x^2}{e^{0.01x}} $ ?

For the functions that you see most commonly, that would be OK. Be careful when the functions vary a lot near $\infty$, such as $(x+\sin(x))/\cos(x)$. But, for simple functions like exponentials and powers, it'll work. — Michael Burr, Mar 06 '15 at 13:13
There is a complete discussion about the circumstances in which you can do this type of substitution here. — rnrstopstraffic, Mar 06 '15 at 16:16

score 0 · Accepted Answer · answered Mar 06 '15 at 13:33

0

The function $n=e^x$ is monotone, so it's justified. In this specific example you don't need to do it: $$ \lim_{n \to \infty} \frac{\log^a n}{n^s} = 0 $$

where $a,s>0$

answered Mar 06 '15 at 13:33

Alex

19,262

I see. So $n(x)$ can be any monotone function? – adeelh Mar 06 '15 at 13:56
yes, I think so. – Alex Mar 06 '15 at 14:00

Simplifying $f(n)$ by substituting, for $n$, an appropriately chosen function $n(x)$ to observe limiting behaviour of $f(n)$. Is this justified?

1 Answers1