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:

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, x^{2} – 25:

**Rewrite 25 as 5**: x^{2}^{2}– 5^{2}: (x^{2}– 5^{2})**Apply the difference of two squares formula from algebra (x**: (x + 5)(x – 5).^{2}– y^{2}= (x + y)(x – y)

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

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:

Which leaves:

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:

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:

- Break into groups: (x
^{2}+ 3x – 4) = (x^{2}– x) + (4x – 4) - Factor out x from x
^{2}– x: x(x – 1) - Factor out 4 from 4x – 4: 4(x – 1)
- Combine the above two expressions: x(x – 1) + 4(x – 1)
- Factor out the common term (x – 1): (x – 1)(x + 4)

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

Step 3: Cancel like terms.

Step 4: Use substitution to solve.

lim_{x→-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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