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Home›Calculus›Limits

The limit of a function

The function (x² − 4)/(x − 2) has no value at x = 2, yet at x = 1.999 it gives 3.999 and at 2.001 it gives 4.001.

What happens near a point

Take f(x) = (x² − 4)/(x − 2). Put in x = 2 and you get 0/0, which is not a number, so f(2) is undefined. Saying only that gives a poor picture. Try values close to 2: f(1.9) = 3.9, f(1.99) = 3.99, f(2.01) = 4.01, f(2.1) = 4.1. From both sides the outputs close in on 4. We write limx→2 f(x) = 4, read as the limit of f(x) as x approaches 2 is 4. The reason is visible after factoring: x² − 4 = (x − 2)(x + 2), so for every x except 2, f(x) = x + 2. The graph is the line y = x + 2 with one point missing.

OpenStax states the intuitive definition this way. Let f be defined on an open interval containing a, except possibly at a itself. If all values of f(x) approach the real number L as x (not equal to a) approaches a, then the limit of f(x) as x approaches a is L. The phrase except possibly at a itself is the important part. The limit describes the values near a, never the value at a. A function can be undefined at a, defined with a different value, or defined with the same value, and the limit can be the same in all three cases.

xf(x)xf(x)
1.93.92.14.1
1.993.992.014.01
1.9993.9992.0014.001

Estimating with a table

The problem-solving strategy in OpenStax is to build two columns of x-values, one approaching a from below and one from above, and read where the outputs go. For limx→0 (sin x)/x, with x in radians: sin(0.1)/0.1 ≈ 0.998334, sin(0.01)/0.01 ≈ 0.999983, and the values for −0.1 and −0.01 are the same because the function is even. The values approach 1, so the limit is 1. Here there is no algebra trick like factoring; the table and the graph are the evidence, and a later section proves it with the squeeze theorem.

Tables can mislead if you trust too few rows. A calculator works with a limited number of digits, and for x extremely close to 2 it can round (x² − 4) and (x − 2) badly. Use several rows, check both sides, and back the table with a graph or with algebra when you can. A graph has the opposite weakness: it shows the trend clearly but gives no exact value. Used together, they make a good estimate; the limit laws in unit 4 turn many estimates into exact answers.

The line y = x + 2 with a hole at (2, 4) and the value f(2) = 1 placed elsewhere. The limit at 2 is still 4.

The figure above makes the same point as the graph in OpenStax's Example 2.6, where g(−1) = 4 but the values near −1 approach 3. Here the function equals x + 2 everywhere except at 2, where someone has defined f(2) = 1. Near 2 the outputs still close in on 4, so the limit is 4, while the value is 1. A limit and a value can disagree. When they agree, the function is called continuous at that point, the subject of unit 5.

BackgroundLimiting processes are old. OpenStax notes that early mathematicians used them to get better and better approximations of the area of a circle. Archimedes, for one, trapped a circle between regular polygons with 96 sides, one inside it and one outside. The formal definition of a limit as it is written today did not appear until the late 19th century. Calculus was in heavy use for nearly two centuries before anyone could say precisely what a limit was, which is a good reason to start, as they did, with tables and pictures.
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f(x) = (x² − 4)/(x − 2). Move x toward 2 from either side, or press Approach. The table lists f(2 ± 0.1) to f(2 ± 0.0001).
  1. f(x) = (x² − 4)/(x − 2). What is f(2.01)?
  2. limx→3 (x² − 9)/(x − 3) = ?
  3. limx→1 (x² − 1)/(x − 1) = ?
  4. Using a table, limx→2 (x³ − 8)/(x − 2) = ? (Try x = 1.999 and 2.001.)
  5. sin(0.1)/0.1 to four decimal places (x in radians)?
  6. limx→0 (sin x)/x = ?
  7. g(x) = x + 2 for x ≠ 2 and g(2) = 1. What is limx→2 g(x)?
    • 4
    • 1
    • It does not exist
    • 2
  8. Does limx→a f(x) depend on the value f(a)?
    • No
    • Yes
    • Only for polynomials
    • Only if f(a) = 0
  9. limx→−1 (x² + 3x + 2)/(x + 1) = ?
  10. Why check x-values on both sides of a?
    • The two sides might approach different numbers
    • It doubles the accuracy of the calculator
    • Only the right side counts
    • The table needs an even number of rows
Key points
  • limx→a f(x) = L: as x gets close to a (x ≠ a), f(x) gets close to L.
  • The value f(a) does not affect the limit.
  • Estimate with a table from both sides and with a graph.
  • 0/0 at a point means more work is needed, often factoring.
Sources for this course (5)
  • OpenStax, Calculus Volume 1 , chapter 2 Limits: 2.1 A Preview of Calculus (tangent problem and area problem; secant slopes through (1, 1) on f(x) = x² with (2, 4) giving 3 and (3/2, 9/4) giving 2.5; rock s(t) = −16t² + 64 with average velocities −15.84 and −16.16 near t = 0.5; three rectangles under x² + 1 on [0, 3] giving 8: openstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculus)
  • 2.2 The Limit of a Function ((x² − 4)/(x − 2) → 4 at 2, (sin x)/x → 1 at 0 from a table, sin(1/x) has no limit at 0, one-sided limits of |x − 2|/(x − 2), infinite limits and vertical asymptotes, the formal definition appearing only in the late 19th century: .../pages/2-2-the-limit-of-a-function)
  • 2.3 The Limit Laws (sum, difference, constant multiple, product, quotient, power and root laws; direct substitution; factoring and rationalizing 0/0 forms; squeeze theorem: .../pages/2-3-the-limit-laws)
  • 2.4 Continuity (three conditions; removable, jump and infinite discontinuities; Intermediate Value Theorem: .../pages/2-4-continuity)
  • 2.5 The Precise Definition of a Limit (.../pages/2-5-the-precise-definition-of-a-limit).

Checked: 2026-10-17