• ${x \to a}$ 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 or 2.999.


• ${x \to a^-}$ describes what happens when $x$ is slightly less than $a$, ignoring what happens when $x$ is slightly greater than $a$.

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):

• $\lim f(x) = \ell$ means that $f(x)$ is close to the number $\ell$. This is the most common type of limit.


• $\lim f(x) = \infty$ means that $f(x)$ grows without bound, 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.


• $\lim f(x) = -\infty$ means that $f(x)$ goes extremely negative and never comes back, eventually becoming less than any number (say, minus a trillion) that you care to name.