• ${x \to a}$ 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.