The notation \( \lim_{x \to c^-} f(x) \) is read as
“the limit of \(f(x)\) as \(x\) approaches \(c\) from the left.”
It denotes the number the that outputs \(f(x)\) get closer and closer to
as the input \(x\) approaches \(c\) from the left (if that number exists!).
Similarly \( \lim_{x \to c^+} f(x) \) is read as
“the limit of \(f(x)\) as \(x\) approaches \(c\) from the right,”
and denotes the number the that outputs \(f(x)\) get closer and closer to
as the input \(x\) approaches \(c\) from the right.
If both of these one-sided limits exist and are the same number, say the number \(L,\)
then we drop the superscript \(+/-\) in the notation and simply write
\[ L = \lim\limits_{x \to c} f(x)\,. \]
If \(f\) has a pole at \(c,\) then the graph of \(f\) will have a vertical asymptote of \(x=c.\) In this case, while technically \(\lim_{x \to c} f(x)\) doesn’t exist, we often abuse this limit notation and write \(\lim_{x \to c} f(x) = \infty\) or \(\lim_{x \to c} f(x) = -\infty\) if \(f\) behaves uniformly around the pole.
This limit notation is often also overloaded to describe the end behavior of a function.
The limit \(\lim_{x \to \infty} f(x)\) denotes the number
that the output \(f(x)\) approaches as \(x\) increases to \(\infty,\) if that number exists.
Similarly \(\lim_{x \to -\infty} f(x)\) denotes the number
that the output \(f(x)\) approaches as \(x\) decreases to \(-\infty.\)
These limits only exist if the graph of \(f\) has a horizontal asymptote,
and eventually “levels-out” on the right or left.