One-Sided Limits (AP Exam Favorite)

Builds directly on the limit notation from the previous video by adding the small $-$ and $+$ superscripts.

Reading the notation

$\lim_{x \to 2^-} f(x) \qquad \text{vs} \qquad \lim_{x \to 2^+} f(x)$

• The minus means approaching from the left — from $x$ values less than $2$.
• The plus means approaching from the right — from $x$ values greater than $2$.

The video notes both phrasings mean the same thing; pick whichever you find easier to remember and stick with it.

When does the limit exist?

$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

("iff" is just mathematician shorthand for if and only if.) So you must check both sides before you can claim the two-sided limit exists.

Worked example

Two graphs are read off the board, $f(x)$ and $g(x)$, each asking for the limit as $x \to 2$.

For $f(x)$: approaching $2$ from the left, the $y$-values close in on $4$; approaching from the right, they also close in on $4$. The one-sided limits agree, so

$\lim_{x \to 2} f(x) = 4$

For $g(x)$: approaching from the left, the $y$-values climb towards $2$ (the graph shows an open circle there — $g$ is not defined at that point, but the limit only cares about the journey). Approaching from the right, the graph jumps to a visibly higher branch and the $y$-values head towards a different, larger value. Since

$\lim_{x \to 2^-} g(x) \neq \lim_{x \to 2^+} g(x)$

the two-sided limit does not exist, written DNE on the board.