edubun
English
English한국어EspañolPortuguêsFrançaisDeutschItalianoРусскийBahasa Indonesia日本語
Home›Calculus›Limits

The limit laws

Put x = 9 into (√x − 3)/(x − 9) and you get 0/0. Multiply top and bottom by √x + 3 and the limit turns out to be exactly 1/6.

From estimates to exact values

Tables give good guesses, but they never prove anything. The limit laws let you compute limits exactly by breaking a function into simple pieces. Two basic limits start everything: limx→a x = a, and limx→a c = c for a constant c. Both are obvious from a graph. The laws then say how limits combine. Suppose limx→a f(x) = L and limx→a g(x) = M, both real numbers.

  • Sum and difference: lim (f ± g) = L ± M.
  • Constant multiple: lim c·f = c·L.
  • Product: lim f·g = L·M.
  • Quotient: lim f/g = L/M, provided M ≠ 0.
  • Power: lim fn = Ln for a positive integer n.
  • Root: lim n√f = n√L (for even n, L must be at least 0).

Applied one after another, these laws show that for any polynomial p, limx→a p(x) = p(a). The same holds for a rational function p(x)/q(x) wherever q(a) ≠ 0. So most of the time, if a function is built from x with sums, products and quotients, and the denominator is not zero at a, you can just substitute. That is direct substitution. For example, limx→3 (2x² − 3x + 1) = 18 − 9 + 1 = 10, and limx→2 (x² + 1)/(x + 3) = 5/5 = 1.

When substitution gives 0/0

The interesting limits are the ones where substitution fails. If both top and bottom go to 0, the quotient law does not apply, and 0/0 tells you nothing yet: the limit could be any number, or not exist. The fix is to rewrite the function so that it agrees with a nicer one for x ≠ a. Factoring is the first tool. For (x² − 5x + 6)/(x − 3), the top is (x − 2)(x − 3), so for x ≠ 3 the function equals x − 2 and the limit at 3 is 1. Canceling is legal because the limit never uses x = 3 itself.

The second tool is multiplying by a conjugate when a square root is involved. For (√x − 3)/(x − 9), multiply the top and bottom by √x + 3. The top becomes x − 9, which cancels with the bottom and leaves 1/(√x + 3). Now substitution works: 1/(3 + 3) = 1/6. A third tool, for complex fractions, is to combine the small fractions over a common denominator first. All three tools do the same job: they remove the factor that makes both parts zero.

The squeeze theorem

Some limits resist algebra. The squeeze theorem handles them: if f(x) ≤ g(x) ≤ h(x) for x near a, and f and h both have limit L at a, then g has limit L too. It is trapped. OpenStax uses it with a picture of the unit circle to prove that (sin θ)/θ approaches 1 as θ approaches 0, the limit unit 2 could only estimate with a table. That limit becomes the key step when the derivative of sine is worked out in the derivatives course.

Before you factor anything, try substitution first. If it gives a real number with a nonzero denominator, that number is the limit and you are done. Only a 0/0 result calls for the rewriting tools above, and a nonzero number over 0 points to an infinite limit instead.
Loading

Drag to turn. Scroll or pinch to zoom.

(sin x)/x near 0. Approach from either side; the squeeze theorem proves the value the table shows, 1.
  1. limx→3 (2x² − 3x + 1) = ?
  2. limx→2 (x² + 1)/(x + 3) = ?
  3. limx→3 (x² − 5x + 6)/(x − 3) = ?
  4. limx→9 (√x − 3)/(x − 9) = ? (decimal, 3 places)
  5. limx→−1 (x² − 1)/(x + 1) = ?
  6. lim f = 4 and lim g = −2 at a. lim (3f − g) = ?
  7. Same f and g: lim f·g = ?
  8. Same f and g: lim f/g = ?
  9. Same f: lim f³ = ?
  10. Substituting gives 0/0. What does that tell you about the limit?
    • Nothing yet; rewrite the function
    • The limit is 0
    • The limit does not exist
    • The limit is 1
  11. If 1 − x² ≤ g(x) ≤ 1 + x² near 0, then limx→0 g(x) is:
    • 1
    • 0
    • 2
    • unknown
Key points
  • Limit laws: sums, multiples, products, quotients (nonzero denominator), powers and roots.
  • Polynomials and rational functions with q(a) ≠ 0: substitute directly.
  • 0/0: factor, multiply by a conjugate, or combine fractions, then substitute.
  • Squeeze theorem: a function trapped between two with the same limit shares it.
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