Estimating Limits from Graphs & Tables
A limit describes the value a function approaches as the input gets closer to a point — it does not depend on the function's actual value (or existence) at that point.
What a limit actually asks
Writing lim(x→c) f(x) = L means: as x gets arbitrarily close to c (from either side), f(x) gets arbitrarily close to L. The limit is about the trend approaching c, not the value at c — f(c) could be different, or undefined, and the limit can still exist.
To read a limit off a graph, cover up the point x = c with your finger and trace the curve from the left and from the right. If both sides walk into the same height, that height is the limit. If they walk toward different heights, the limit does not exist (DNE) at that point.
From a table, look at the pattern of f(x) values as x gets closer to c from below and from above. If both sequences converge to the same number, that's the limit.
Worked Example
A table shows f(1.9)=3.8, f(1.99)=3.98, f(1.999)=3.998 and f(2.1)=4.2, f(2.01)=4.02, f(2.001)=4.002. Find lim(x→2) f(x).
- From the left (x < 2), values are approaching 4 (3.8, 3.98, 3.998 → 4).
- From the right (x > 2), values are also approaching 4 (4.2, 4.02, 4.002 → 4).
- Both one-sided trends agree on the same target value.
Answer: lim(x→2) f(x) = 4
One-sided limits and when a limit fails to exist
lim(x→c⁻) f(x) is the left-hand limit (x approaching c from values less than c); lim(x→c⁺) f(x) is the right-hand limit. The two-sided limit lim(x→c) f(x) exists only when both one-sided limits exist AND are equal.
Common reasons a limit doesn't exist: the two sides approach different heights (a jump discontinuity), the function oscillates infinitely near c, or the function grows without bound (in which case we say the limit is ∞ or -∞, which is technically still 'does not exist' as a finite number, but we describe the unbounded behavior explicitly).
Key terms (3)
- Limit
- The value a function approaches as the input approaches a given point.
- One-sided limit
- The value approached from only the left (⁻) or only the right (⁺) of a point.
- DNE
- 'Does not exist' — shorthand used when a limit has no single finite value.
Practice Quiz
The graph of f has a filled dot at (3, 5) and the curve on both sides of x = 3 approaches the open circle at (3, 2). What is lim(x→3) f(x)?