1.3 Estimating Limit Values from Graphs

Reading limits from a graph

To estimate lim(x→c) f(x) from a graph, trace the curve from both the left and right sides toward x = c and see what y-value the graph approaches — NOT the actual value at x = c. The function's value at c (if it even exists, shown by a filled or open dot) doesn't have to match the limit.

One-sided limits and existence

The left-hand limit lim(x→c⁻) f(x) looks at behavior approaching from the left; the right-hand limit lim(x→c⁺) f(x) looks from the right. The overall limit exists only if both one-sided limits are equal. Graphs with jumps, holes, or asymptotes reveal these behaviors clearly.

Common graph features to watch for

Jump discontinuities (branches meet at different heights) mean the limit doesn't exist. Removable discontinuities (open circle) still have a limit equal to where the hole 'should' be. Vertical asymptotes cause the limit to approach ±∞, so it does not exist as a finite value.

Key terms

Two-sided limit existence

lim(x→c) f(x) exists only if lim(x→c⁻) f(x) = lim(x→c⁺) f(x).

Jump discontinuity

Left and right branches meet at different y-values at x = c; the limit does not exist there.

Removable discontinuity

An open circle 'hole' in the graph; the limit still exists and equals the height the graph approaches.

Vertical asymptote and limits

As x approaches the asymptote, f(x) → ±∞, so the limit does not exist (it's unbounded).

Estimating a limit from a graph

Trace the curve from both sides toward x = c and find the y-value it approaches — ignore the actual function value at c.

Try an AP-style question

What is lim(x→2) f(x)?

1. 3
2. 4
3. Does not exist
4. 5