Graphical Limits: Interactive Guide
To read a limit from a graph, trace the curve toward the target x-value from each side; the limit equals the single height both sides approach, even if the plotted point at that x-value is somewhere else, and it does not exist if the two sides approach different heights.
The easiest way to think about limits is to imagine you're on a roller coaster riding the track from one side or the other:
Left-Hand Limit (x ➔ c⁻)
• What it means: Tracing the track coming from the left-hand side (indicated by the negative exponent superscript).
• In the simulation: Drag the red roller coaster cart from the left toward the target x-value (e.g., x = 5). As it gets closer, it turns blue, showing you the height (y-value) f(x) approaches from the left-hand side.
What is the rule for finding a limit from a graph?
Graphical limit worked example: a hole versus a point
Trace a piecewise graph to determine the limit of f(x) as x approaches 2, where f(x) has a hole at (2, 3) and a point at (2, 1).
1. Tracing the curve from the left: as x approaches 2 from values less than 2, the y-value approaches 3.
2. Tracing the curve from the right: as x approaches 2 from values greater than 2, the y-value also approaches 3.
3. Because both one-sided limits match:
Note that the actual value of the function f(2) = 1 does not affect the limit.
Another worked example
Find the limit as x approaches 4 from the left for a graph that rises to an open circle at height 2 from the left side. Step 1: trace the curve only from x-values less than 4. Step 2: read the height the curve approaches, which is 2. Step 3: the left-hand limit equals 2, regardless of any separate point plotted at x = 4.
Aligned with the College Board AP Calculus CED — Unit 1 (Limits and Continuity), Topic 1.7, with the graphical estimation of Topics 1.3 and 1.4. College Board ↗
Interactive simulator is loading...
Engaged with the simulation?
Practice actual exam problems and master calculus guaranteed. Get a 5 on the AP Calculus exam or get an A in the calculus class, or we'll pay you $100.