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Daily Math Minute

Limits & Continuity

Estimating Limits

Estimating a limit from a table of values or a graph.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

What a Function Approaches Near a Point

f(x) = (x² − 9)/(x − 3) is undefined at x = 3 — direct substitution gives 0/0. Before reading on: does that mean nothing can be said about how f behaves right around x = 3?

Definition — Limit

limₓ→c f(x) is the value f(x) approaches as x gets arbitrarily close to c from both sides — independent of whether f is even defined at c. A two-sided limit exists only when the left-hand and right-hand limits agree.

Worked Example — Estimating a Limit Numerically

Estimate limₓ→3 (x² − 9)/(x − 3) using a table approaching x = 3 from both sides. From the left: f(2.9) = 5.9, f(2.99) = 5.99. From the right: f(3.1) = 6.1, f(3.01) = 6.01. Both sides close in on 6, so limₓ→3 f(x) = 6 — matching the algebraic simplification (x−3)(x+3)/(x−3) = x+3, which equals 6 at x = 3.

Function Explorer

Transform: g(x) = a·f(b(x − h)) + k
1
1
0
0
Composition
Analysis (of the transformed function, in view)
y-intercept
(0, 0)
x-intercepts
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema
local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points
(-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [-1, 1]

Worked Example — One-Sided Limits That Disagree

Let g(x) = 2x for x < 2, and g(x) = x + 5 for x ≥ 2. As x → 2⁻: g(x) → 2(2) = 4. As x → 2⁺: g(x) → 2 + 5 = 7. Since the left-hand limit (4) and right-hand limit (7) disagree, limₓ→2 g(x) does not exist — even though g(2) = 7 is a perfectly defined value.

Tip

When direct substitution gives a defined number, that's usually the limit. When it gives 0/0, don't stop — a table of values or algebraic simplification can still reveal a real limit.

Common Mistakes

  • Assuming a limit doesn't exist just because the function is undefined at that exact point.

    A function being undefined at c says nothing by itself about whether limₓ→c f(x) exists — (x² − 9)/(x − 3) is undefined at x = 3 but still has a limit of 6 there.

  • Checking values from only one side and concluding the two-sided limit exists.

    A genuine two-sided limit requires both the left-hand and right-hand limits to exist and agree — always check both directions.

Key Takeaways

  • A limit describes what a function approaches near a point, independent of the function's actual value there.
  • A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
  • Tables of values approaching from both sides can estimate a limit when direct substitution fails.

Summary

Estimating limits builds the intuition; the next lesson develops algebraic techniques to evaluate them exactly.

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