Solving Limits by Factoring

Calculus 1 · Axiom Academy

EXAMPLE Solving Limits by Factoring When direct substitution gives the indeterminate form 0/0, factor and cancel to reveal the limit. Evaluate the limit . Plugging in x = 2 fails, so we will need a way to simplify the expression first. Nice work. You resolved a 0/0 limit by factoring out the term that caused it — the core technique for limits of rational functions. Spot the indeterminate form: direct substitution giving is a signal to simplify, not the final answer. Factor, then cancel: the numerator x^2 - 4 = (x+2)(x-2) shares the factor x - 2 with the denominator, which is what creates the . Cancelling is legal here: a limit asks what happens as , where , so dividing by x - 2 is valid. The simplified function x + 2 agrees with the original everywhere except at x = 2 , where the original has a hole (a removable discontinuity) — yet the limit still exists and equals 4.

This is the written version of the interactive lesson above. See the full Calculus 1 course.