AP Calculus BC / Unit 1: Limits and Continuity
1.3 Estimating Limit Values from Graphs
Notes
To read a limit from a graph, trace the curve toward $x=c$ from each side and see what $y$-value it approaches. Open circles (holes) and filled dots can disagree with the limit.
- A hole at $(c, L)$ still has limit $L$.
- A jump means the one-sided limits differ, so the two-sided limit doesn't exist.
- A vertical asymptote means the function grows without bound (an infinite limit).
Practice
A graph has an open circle at $(3, 2)$ and a filled dot at $(3, 5)$, with the curve approaching 2 from both sides. What is $\displaystyle \lim_{x\to 3} f(x)$?
The limit follows the curve, not the isolated point.
For that same graph, what is $f(3)$?
The filled dot shows the function's actual value.
A graph jumps from height 1 (left of $x=0$) to height 4 (right of $x=0$). What is $\displaystyle \lim_{x\to0} f(x)$?
The left and right limits are different.
Ready for more? Take the Unit 1 test.