Is it correct to express $\lim \limits_{\eta\to\infty}f(\eta)$ as $f(\infty)$?

Question

Hi I have a really basic question that unclear to me for long long time...

Is it correct to express $\lim \limits_{\eta\to\infty}f(\eta)$ as $f(\infty)$?

I always use the former, however I saw the latter used very often but not sure they are the same thing.

Any advice is appreciated!

Answer 1

If $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta ) $ exists then $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta ) $ is either finite or infinite.

If $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta ) $ is finite then $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=l $ for some $ l\in \mathbb{R} $.

Then $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=l $ means for any $ \epsilon >0 $, there exists $ a_{\epsilon}\in \mathbb{R} $ such that for each $ \eta >a_{\epsilon} $, $ \vert f(\eta )-l\vert <\epsilon $.

If $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta ) $ is infinite then either $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=\infty $ or $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=-\infty $.

$ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=\infty $ means for any $ M>0 $, there exists $ a_{M}\in \mathbb{R} $ such that for each $ \eta >a_{M} $, $ f(\eta )>M $.

Similarly $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta )=-\infty $ means for any $ M>0 $, there exists $ a_{M}\in \mathbb{R} $ such that for each $ \eta >a_{M} $, $ f(\eta )<-M $.

But $ f(\infty ) $ does not mean anything. Therefore you can't express $ \mathop {\lim }\limits_{\eta \to \infty } f(\eta ) $ as $ f(\infty ) $.