It may help to list these "different kinds of limits", each with their own formal definition and notation, to get an idea of what is going on and why certain terminilogy about (not) existing is used.
$x$ tends to real, limit is real.
Usually, one begins with defining the situation where, if $x$ tends to $a$ where $a$ is a real number, the function tends to a real number $L$ which we call the limit of $f$ at $a$ and we write this as:
$$\lim_{x \to a} f(x) = L$$
When there is no real number $L$ satisfying the formal definition (which I left out), then you say the limit does not exist. This is the "basic case" but it is useful to extend the use of limits to be able to describe other situations.
$x$ tends to real, limit is infinite.
One of the reasons why the limit may not exist, is because the function grows arbitrarily large (or small; negative) when $x$ tends to $a$. This could be useful information which is lost when we would just say that the limit does not exist. To capture this information, we agree to write this as:
$$\lim_{x \to a} f(x) = +\infty \quad (\mbox{or}\; -\infty)$$
So in fact we introduce a notation that symbolizes this behavior, but there are reasons to still say that "the limit does not exist". We reserve this for the case where the limit is a real number because with this choice, a lot of properties of limits are more elegant and/or easier to state.
$x$ tends to infinity, limit is real.
Limits are a way to describe the behaviour of function where its value cannot be simply evaluated. It is interesting to be able to describe how a function behaves when $x$ grows arbitrarily large (or small; negative). If a function tends to a real number $L$ when $x$ grows arbitrarily large, we say that the limit of $f$ at (positive) infinity exists and we write:
$$\lim_{x \to +\infty} f(x) = L$$
This is a different notation and describes a different situation. The "role of infinity" is not the same: it is not $|f|$ that grows arbitrarily large but rather we choose to let $x$ grow arbitrarily large and introduce a new notation for this. The limit can exist (with the notation above), or not.
$x$ tends to infinity, limit is infinite.
Again, if the limit in the situation above does not exist in the sense that there is no real number $L$ satisfying its (omitted) formal definition, there are different kinds of possible reasons. For example, the sine function does not tend to a fixed real number so the following limit "does not exist":
$$\lim_{x \to +\infty} \sin x $$
It is however possible that, like in case #2, the function grows arbitrarily large (or small; negative) when we let $x$ grow arbitrarily large (or small; negative). We capture this behaviour by introducing the notations:
$$\lim_{x \to \pm\infty} f(x) = \pm\infty$$
where I used $\pm$ twice to briefly summarize the four possibilities.
Sometimes, when all these cases are in a sense "allowed", text books will explicitly state that in the notation
$$\lim_{x \to a} f(x) = L$$
they allow the numbers $a$ and/or $L$ to be $\pm \infty$ as well so they don't have to run down all the different notations/cases. Keep in mind though that these different types of limits all have their own formal definitions.