Can I say a limit has a value of $\infty$? Can I say that limits like $\lim_{x\to\infty}x$ are equal to infinity?
E.g. the limit as $x\to+\infty$ of $x^2$ is infinity.
Is this a correct statement?
Sometimes my teacher says the above, sometimes they'd use the terms "inderterminate" or "undefined" also. 
I wonder which is correct. Is it correct to say it's undefined?
 A: Yes, you can, provided that the right definition is given.
The notation
$$\lim_{x\to a}f(x)=\infty$$
is very common and acceptable. Well, the symbol $\infty$ is read as "infinity" so it is natural to say that a limit like that is infinite. You only have to know what it means exactly and make it clear for the audience (students, readers, etc.).
The key point is to give a precise meaning for your terminology. So, if you want call these limits "infinite", just put it in the definition. See for example how Apostol treat this topic:

A: Remember what limits mean. When I say that $f(x) \to f(c)$ as $x \to c$, I'm saying that I can make $f(x)$ as close to $f(c)$ as I want by making $x$ very close to $c$. When I say $f(x) \to c$ as $x \to \infty$ I'm saying that $f(x)$ can be made very close to $c$ by choosing $x$ sufficiently large.
An infinite limit means that as $x \to c$, $f(x)$ become arbitrarily large. I can make $f(x)$ as large as I want by choose $x$ sufficiently close to $c$. The notation $\lim_{x \to c} f(x) = \infty$ conveys this behavior and it's certainly well defined. Likewise, $f(x) \to \infty$ as $x \to \infty$ means that as $x$ grows arbitrarily large, $f(x)$ becomes arbitrarily large. I can always make $f(x)$ larger by making $x$ larger. The notation $\lim_{x \to \infty} f(x) = \infty$ conveys this behavior and it's certainly well defined.
However, this notation does not treat $\infty$ as a number, but as a descriptor of behavior.
The reason why we don't say that $\lim_{x \to 0} \frac{1}{x} = \infty$ is that the behavior of $\frac{1}{x}$ depends on the direction we approach $0$; approaching from the right makes this very large while approaching from the left makes it very small (negative numbers are small). Thus the limit is not defined.
A: Much of single variable calculus is most naturally set in the extended real number line. (and most of what is not is best set in the real projective line)


*

*When the value of $x$ is taken to be in the reals, $\lim_{x \to +\infty} x^2$ limit is undefined.

*When the value of $x$ is taken to be in the extended reals, $\lim_{x \to +\infty} x^2$ exists and is equal to $+\infty$.


Note that in either case the domain of $x$ is taken to be in the extended reals, so that it makes sense to talk about $x \to \infty$.
