The limit of a function as x--> infinity

[Matejxx1]

Hi everyone,
So we were writting our math test today and I am not completely sure about one concept.
For the sake of simplicity let's say that
f(x)=x²
and let's say we were asked to find,
lim f(x) as x--->infinity = ?
is the correct answer here undefined or infinity.
Thanks for the help

 

[fresh_42]

Both to some extend. Strictly speaking the limit is undefined because there is no number $f(x)$ converges to. And $∞$ cannot be used within the formal definition of a limit. However, in contrast to a situation like, e.g. $\lim_{n→∞}{(-1)^n}$ we may say that the limit increases beyond all boundaries which is basically the formal definition in case $f(x)$ goes to infinity. Therefore we may note $\lim_{x→∞}{f(x)} = ∞$ for short which indicates the need to apply the "beyond all boundaries" version of the definition and allows us to distinguish such a behaviour from divergent cases as in my example.

 

[HallsofIvy]

Saying that "the limit is infinity" is just shorthand for "the limit does not exist- for a specific reason". That specific reason is that the number get larger and larger without bound. It is also possible that the numbers get lower and lower (are negative number with absolute value getting larger and larger without bound). In that case the limit also does not exist but we might say "the limit is negative infinity". Neither of those is ambiguous because "infinity" and "negative infinity" are not real numbers. There is, of course, a third possibility- that the numbers do not get "larger and larger without bound" nor "lower and lower without bound" but still do not converge. For example $(-1)^n$ or $(-1)^nn$