Skip to content
Glacius
CalculusConcept reference

Limit algebra

Algebraic limit laws evaluate combinations of functions with known finite limits, with a nonzero denominator limit required for a quotient.

1 min read1 figureSources & further reading
On this page 9 sections
  1. Overview
  2. Understand the idea
  3. A quotient needs one more check
  4. For (x^2-2x)/(x-2), factor the numerator as x(x-2)
  5. For polynomials, use xto a and constant limits to substitute a
  6. Key takeaway
  7. Further questions
  8. Sources & further reading
  9. Concept connections

01Understand the idea#

If f(x)→2f(x)\to2 and g(x)→3g(x)\to3 as inputs approach the same target, their sum approaches 5 and their product approaches 6. Both component limits must exist and be finite.

For finite limits F and G at the same input target, sums become F+G, differences F−G, and products FG. A constant multiple becomes cF; a positive whole-number power becomes F raised to that power.

Check your reasoning

As x→2x\to2, f(x)→3f(x)\to3 and g(x)→−1g(x)\to-1. Find lim⁡[2f(x)+g(x)]\lim[2f(x)+g(x)].

Show answer and explanation
5

Apply the laws: 2·3−1=5.

02A quotient needs one more check#

A quotient needs one more check. If the numerator approaches 6 and the denominator approaches 2, the quotient approaches 3. The denominator limit must be nonzero before dividing.

lim⁡(f/g)=6/2=3\lim(f/g)=6/2=3
Check your reasoning

As x→1x\to1, f(x)→4f(x)\to4 and g(x)→2g(x)\to2. Find lim⁡f(x)/g(x)\lim f(x)/g(x).

  1. A8
  2. B2
  3. C0.5
Show answer and explanation
2

The denominator limit is nonzero.

03For (x^2-2x)/(x-2), factor the numerator as x(x-2)#

For (x2−2x)/(x−2)(x^2-2x)/(x-2), factor the numerator as x(x−2)x(x-2). For x≠2, cancelling gives x. Its limit at 2 is 2; the original endpoint stays undefined.

The graph is the line y=x with the point (2,2) removed. It represents (x squared minus 2x)/(x minus 2) for x not equal to 2. Cancelling the common factor preserves every plotted nearby value but does not define a value at x=2. Both axes use 32 pixels per unit.The graph is the line y=x with the point (2,2) removed. It represents (x squared minus 2x)/(x minus 2) for x not equal to 2. Cancelling the common factor preserves every plotted nearby value but does not define a value at x=2. Both axes use 32 pixels per unit.
Figure 1The graph is the line y=x with the point (2,2) removed. It represents (x squared minus 2x)/(x minus 2) for x not equal to 2. Cancelling the common factor preserves every plotted nearby value but does not define a value at x=2. Both axes use 32 pixels per unit.
Link to this figure ↗Download SVGDownload PNG
Check your reasoning

A learner reports 0 for 0/0. Correct lim⁡x→4(x2−4x)/(x−4)\lim_{x\to4}(x^2-4x)/(x-4).

Show answer and explanation
4

For x≠4, the quotient equals x.

04For polynomials, use xto a and constant limits to substitute a#

For polynomials, use x→ax\to a and constant limits to substitute a. Rational functions also allow substitution when their denominator is nonzero at a. A 0/0 form alone gives no answer; simplify a common factor if possible.

Key takeaway

Check the law’s conditions before substituting limits.

  • Evaluate a finite limit using valid algebraic limit laws

Further questions

Does 0/0 mean a limit is zero?
No. It means direct division of the component limits is not valid. The quotient might approach a finite value, fail to approach one, or grow without bound. Further information about the functions is needed.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Limit algebra.” Math behind ML. /learn/c-limit-algebra