Evaluate a Limit by Factoring

Evaluate a Limit by Factoring

When a limit gives a direct substitution result like 0/0, the expression is not yet simplified enough. Factoring often removes the shared factor that causes the problem.

1) Try direct substitution first

Plug the approaching value into the expression. If you get a number, that is usually the limit. If you get an indeterminate form such as 0/0, continue.

2) Factor the numerator and/or denominator

Look for common factors, especially factors like (x-a) or polynomial factors that can be simplified. Rewrite the expression in factored form.

3) Cancel the common factor

If the same factor appears in the numerator and denominator, cancel it carefully. This is allowed only after factoring, not by canceling terms that are added or subtracted.

4) Substitute again

Now evaluate the simplified expression at the target value. Since the troublesome factor has been removed, the limit should be straightforward.

5) Check your result

Make sure the final answer is simplified and that your cancellation was valid. If the original expression was a quotient, verify that the final value matches the simplified form near the target point.

Key idea: factoring turns an undefined-looking expression into an equivalent simpler one, making the limit visible.