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Algebraic limits

Limits by Factoring: Remove a Hole Correctly

A 0/0 substitution is not a final limit. For polynomial quotients it often signals a removable common factor.

Direct answer

Factor, cancel the common nonzero factor near the point, then evaluate.

Cancellation is valid only for nearby x where the canceled factor is nonzero.

When to use limits by factoring

Choose the method from the top-level mathematical structure before manipulating symbols. Conditions are part of the task, not optional notes.

Ask first 1

Which structure controls the method?

What to do

Try factoring when numerator and denominator are polynomials that both vanish at the target.

Ask first 2

Which condition can change the answer?

What to do

Cancellation is valid only for nearby x where the canceled factor is nonzero.

Ask first 3

How should the result be checked?

What to do

State the exclusion used during cancellation and compare both directions.

Use a punctured neighborhood

For (x²−1)/(x−1), direct substitution at 1 gives 0/0.

Factor x²−1=(x−1)(x+1).

For x≠1 the quotient equals x+1, which has the same nearby values.

Evaluate x+1 at 1 to get 2; the original quotient remains undefined there.

1
x2−1x−1\frac{x^{2} - 1}{x - 1}
2
(x−1)(x+1)/(x−1)=x+1 for x≠1
3
lim x→1 =2

Worked examples

lim x→1 (x²−1)/(x−1)
22

Cancel only for x≠1.

lim x→2 (x²−4)/(x−2)
44

The simplified form is x+2.

lim x→3 (x²−5x+6)/(x−3)
11

Factor as (x−2)(x−3).

Common mistakes

  • • Calling 0/0 the answer.
  • • Canceling across addition.
  • • Redefining the original point silently.
  • • Ignoring directional behavior.

Error clinic: locate the first invalid step

Incorrect attempt

limit=0/0

Why it fails

This is an indeterminate substitution result.

Correction

Factor to obtain 2.

Incorrect attempt

x2+1x=x+1\frac{x^{2} + 1}{x} = x + 1

Why it fails

x is not a factor of the full numerator.

Correction

No cancellation is valid.

Practice the same idea with a changed structure

Name the rule and its conditions before revealing the reference answer. Explain every sign and factor.

lim x→2 (x²−4)/(x−2)

Check before solving

The simplified form is x+2.

Reveal the reference answer

Reference answer

44
lim x→3 (x²−5x+6)/(x−3)

Check before solving

Factor as (x−2)(x−3).

Reveal the reference answer

Reference answer

11

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Limites:x2−1x−1\frac{x^{2} - 1}{x - 1}Limites:x2−4x−2\frac{x^{2} - 4}{x - 2}
Open the calculator without a preset: Limites

Frequently asked questions

Does 0/0 mean zero?+

No. It is an indeterminate form requiring more analysis.

Why can the factor be canceled?+

The limit uses nearby values where that factor is nonzero.

Can terms cancel across addition?+

No. Cancellation applies to common factors of the entire numerator and denominator.

Conclusion

When substitution gives 0/0, factor, cancel only genuine factors nearby, and keep the point value separate.

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