Concept preview

What you will be able to do

1. 01Read a two-sided limit from a graph by tracing the curve toward the target from the left and the right.
2. 02Determine one-sided limits and use their agreement (or disagreement) to decide whether the two-sided limit exists.
3. 03Distinguish the limit from the function’s value at the point, including at holes, jumps, and vertical asymptotes.

To find $\lim_{x\to c}f(x)$ from a graph, use a two-finger routine. Place one finger on the curve to the LEFT of $x=c$ and slide it rightward toward $c$, noting the height it approaches — that height is the left-hand limit $\lim_{x\to c^-}f(x)$. Place the other finger to the RIGHT of $c$ and slide leftward toward $c$ for the right-hand limit $\lim_{x\to c^+}f(x)$. If both fingers head to the same height $L$, then $\lim_{x\to c}f(x)=L$. If they head to different heights, the two-sided limit does not exist.