Unit 1: Limits & Continuity

Estimating Limits

Estimating limit values from graphs and tables.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

What a Function Approaches, Not What It Equals

Consider f(x) = (x² − 1)/(x − 1). Plug in x = 1 directly and you get 0/0 — undefined. Before reading on: does that mean nothing useful can be said about f near x = 1, or can you learn something about f's behavior there without ever evaluating f(1) itself?

Definition — Limit

The limit of f(x) as x approaches c, written limₓ→c f(x), is the value f(x) gets arbitrarily close to as x gets arbitrarily close to c from both sides — regardless of whether f is even defined at c, and regardless of what f(c) equals if it is defined.

\lim_{x \to c} f(x) = L

Worked Example — Estimating a Limit Numerically

Estimate limₓ→1 (x² − 1)/(x − 1) using a table of values approaching x = 1 from both sides. From the left: f(0.9) = (0.81 − 1)/(0.9 − 1) = 1.9, f(0.99) = 1.99, f(0.999) = 1.999. From the right: f(1.1) = (1.21 − 1)/(1.1 − 1) = 2.1, f(1.01) = 2.01, f(1.001) = 2.001. Both sides close in on 2, so limₓ→1 f(x) = 2 — even though f(1) itself is undefined. This matches simplifying algebraically: (x² − 1)/(x − 1) = (x − 1)(x + 1)/(x − 1) = x + 1 for x ≠ 1, and x + 1 evaluated at x = 1 is 2.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

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]

A limit only exists when the function approaches the same value from both directions. Approaching from x < c gives the left-hand limit; approaching from x > c gives the right-hand limit. When those two one-sided limits disagree, the two-sided limit does not exist (DNE) at that point — even if the function has a perfectly well-defined value there.

Worked Example — One-Sided Limits That Disagree

Let g(x) = x² for x < 2, and g(x) = x + 3 for x ≥ 2. As x → 2⁻ (from the left, using the x² piece): g(x) → 2² = 4. As x → 2⁺ (from the right, using the x + 3 piece): g(x) → 2 + 3 = 5. Since the left-hand limit (4) and right-hand limit (5) disagree, limₓ→2 g(x) does not exist — even though g(2) = 2 + 3 = 5 is a perfectly defined value. A function's value at a point and its limit at that point are two separate questions.

Tip

When direct substitution gives a defined number, that number usually is the limit. When it gives 0/0 or another undefined form, don't stop — a table of values approaching from both sides, or an 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 − 1)/(x − 1) is undefined at x = 1 but still has a limit of 2 there.
Checking values approaching from only one side and concluding the two-sided limit exists.
A genuine two-sided limit requires the left-hand and right-hand limits to both exist and agree — always check both directions before declaring a limit exists.

Key Takeaways

A limit describes what a function approaches near a point, independent of (and sometimes despite) 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, and algebraic simplification, are two ways to estimate or find a limit when direct substitution fails.

Summary

Estimating limits numerically and graphically builds the intuition; the next lesson develops the algebraic techniques and limit laws that evaluate limits exactly, without needing a table.

Track your progress

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