The most general
limit statement is $$\lim_{x \to \tiny\hbox{something}} f(x)
= \hbox{something else}.$$ Here is what $x$ can do:
 ${x \to a}$ describes what happens when $x$ is close
to, but not equal to, $a$. So $\displaystyle\lim_{x \to
3} f(x)$ involves looking at $x=3.1, 3.01, 3.001,2.9,
2.99, 2.999$, and generally considering all values of
$x$ that are either slightly above or slightly below 3.
 ${x \to a^+}$ describes what happens when $x$ is
slightly greater than $a$. That is,
$\displaystyle\lim_{x \to 3^+}f(x)$ involves looking at
$x=3.1, 3.01, 3.001$, etc.,but not $2.9, 2.99, 2.999$,
etc.
 ${x \to a^}$ describes what happens when $x$ is
slightly less than $a$. That is,
$\displaystyle\lim_{x \to 3^}f(x)$ involves looking at
$x= 2.9, 2.99, 2.999$, etc. and ignoring what
happens when $x=3.1, 3.01, 3.001$, etc.
Note that if something happens as $x \to a^+$ and the
same thing happens as $x \to a^$, then the same also
happens as $x \to a$. Conversely, if something happens as
$x \to a$, then it also happens as $x \to a^+$ and as $x
\to a^$.
Here is what the limit can be (if it exists):
 $\displaystyle\lim_{x\to a} f(x) = L$ means that
$f(x)$ is close to the number $L$ when $x$ is near $a$.
This is the most common type of limit.
 $\displaystyle\lim_{x\to a} f(x) = \infty$ means that
$f(x)$ grows without bound as $x$ approaches $a$,
eventually becoming bigger than any number you can name.
Remember that $\infty$ is not a number! Rather, $\infty$
is a process of growth that never ends.
 $\displaystyle\lim_{x\to a} f(x) = \infty$ means that
as $x$ approaches $a$, $f(x)$ goes extremely negative
and never comes back, eventually becoming less than any
number (say, minus a trillion) that you care to name.
