When is it possible to make a change of variables in the limit?

Question

For example $\lim_{x \to \infty}(\ln x/x)$, can I change $x=e^{y}$?

Then $\lim_{x \to \infty}(\ln x/x)= \lim_{y \to \infty}(y/e^{y})$?

How can I prove that the change of variables is valid?

See theorem mentioned at the end of https://math.stackexchange.com/a/1073047/72031 which can be adapted to deal with situation where we have $x\to\infty$. Your substitution is valid — Paramanand Singh, Oct 27 '18 at 03:32
Such kind of theorems as mentioned in my previous comments are on par with algebra of limits and squeeze theorem and should be used regularly in evaluation of limits. — Paramanand Singh, Oct 27 '18 at 03:35

score 0 · Answer 1 · answered Oct 27 '18 at 03:16

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According to the definition of the limit, $\lim_{x\rightarrow\infty}f(x) = L \iff \forall \epsilon>0, \exists N >0,|f(x)-L|<\epsilon,\forall x>N$.

As long as you can find a transformation $T(x)$ such that if $x>N$, then $T(x)>N$, you sill have that $|f(T(x))-L|<\epsilon$.

In your example, its obvious that if $x>N \implies e^x > N$.

answered Oct 27 '18 at 03:16

Zamarion

Thank you!
And if $\lim_{x \to a}$ ?
– ZAF Oct 27 '18 at 03:24
The definition changes to $\forall \epsilon >0, \exists \delta >0$ such that if $|x-a|<\delta$ then $|f(x) - L|< \epsilon$. Its a bit more tricky to find a $T(x)$ such that if $|x-a|< \delta$, then $|T(x)-a|< \delta$ but it's doable in certain case. – Zamarion Oct 27 '18 at 03:32

score 0 · Answer 2 · answered Oct 27 '18 at 09:38

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Let $x_n > 0.$

$x_n \rightarrow \infty$, then

$y_n: =\log x_n \rightarrow \infty$(why?).

$\lim_{n \rightarrow \infty} \log (x_n)/x_n =$

$\lim_{n \rightarrow \infty} y_n/(e^{y_n})=0$

answered Oct 27 '18 at 09:38

Peter Szilas

2 Answers2