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CalculusBy Educator Today

C02 · Unit C1 Limits and continuity

Limit laws, factoring and rationalization

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 1.5–1.7
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Limit laws allow sums, products and quotients when their hypotheses hold; the quotient law requires a nonzero denominator limit. Direct substitution works for continuous expressions on their domains. A 0 over 0 form calls for analysis, often factoring or rationalization. Cancellation is valid away from the approaching point and can preserve the limit without restoring the original function’s missing value.

Worked example

lim as x tends to 4 of ( square root of x minus 2)/(x minus 4) is found by multiplying by ( square root of x+2)/( square root of x+2). For x is not equal to 4 the expression equals 1/( square root of x+2), whose limit is 1 over 4. The original fraction remains undefined at 4.

Explore the mathematics

Limit laws, factoring and rationalization

Approach a missing point from both sides without substituting into an undefined expression.

Setting 3 of 5

Scroll sideways to see the full diagram.

What value do the outputs approach near x=2?xy1234f(x)=(x²−4)/(x−2), x≠2; h=0.125

Predict before continuing

What value do the outputs approach near x=2?

Read the complete explanation
  1. What value do the outputs approach near x=2?
  2. Approach from both sides; the point at x=2 is missing.
  3. At x=2±0.125, the outputs are 4±0.125.
  4. The limit is 4 although f(2) is undefined. Defining f(2)=4 repairs this removable discontinuity.
See the idea: Limit laws, factoring and rationalization
See the idea: Limit laws, factoring and rationalizationThe simplified curve 1/(√x+2) approaches 1/4 at x=4. The open point is excluded from the original quotient (√x−2)/(x−4), which is undefined at 4.xy0020.1540.360.4580.6excluded (4,¼)

The simplified curve 1/(√x+2) approaches 1/4 at x=4. The open point is excluded from the original quotient (√x−2)/(x−4), which is undefined at 4.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

0 over 0 is an indeterminate form, not the number zero and not proof that the limit does not exist.

Investigate and explain

Compare a removable discontinuity with 1/(x minus 2). Identify why factoring resolves one but not the other.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.