Calculus How To

Factoring limits Examples

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Limit of functions >

Factoring limits is a way to find limits by canceling common factors. Usually, you’ll want to try direct substitution first. If you end up with an indeterminate limit like 0/0, try using factoring to turn the function into one where you can use substitution.

Factoring Limits Examples

Example #1: Find the limit of the following function by factoring:
factoring limits 1

Step 1: Factor the numerator, denominator, or both (this depends on what is factorable in your expression). For this example, we’re factoring the numerator, x2 – 25:

  1. Rewrite 25 as 52: x2 – 52: (x2 – 52)
  2. Apply the difference of two squares formula from algebra (x2 – y2 = (x + y)(x – y): (x + 5)(x – 5).

Step 2: Replace the factored parts of the expression back in the formula:
factor limits

Step 3: Cancel like terms. For this example, we have (x – 5) in the numerator and denominator so we can cancel both of those out:
find limits by factoring

Which leaves:
find limits by factoring 2

Step 4: Use substitution to solve. For this example, plug 5 (the term specified by the “lim” expression) into the formula on the right (x + 5):

(5 + 5) = 10.

The limit is 10.

Example #2: Find the limit of the following function by factoring:
find limits by factoring example 2

Step 1: Factor the numerator, denominator, or both (this depends on what is factorable in your expression). For this example, we can factor the numerator:


  1. Break into groups: (x2 + 3x – 4) = (x2 – x) + (4x – 4)
  2. Factor out x from x2 – x: x(x – 1)
  3. Factor out 4 from 4x – 4: 4(x – 1)
  4. Combine the above two expressions: x(x – 1) + 4(x – 1)
  5. Factor out the common term (x – 1): (x – 1)(x + 4)

Step 2: Replace the factored parts of the expression back in the formula:
factoring limits examples

Step 3: Cancel like terms.
factoring limits examples 2

Step 4: Use substitution to solve.
limx→-4 (x – 1) = -5.


References

7. Limits by algebraic simplification. Retrieved July 14, 2021 from: https://web.auburn.edu/holmerr/1617/Textbook/limbyalg-screen.pdf

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Stephanie Glen. "Factoring limits Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/factoring-limits-examples/
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