Step-by-Step: Solving an Indeterminate Limit

1. Try direct substitution first.
2. If you get 0/0, factor the numerator and denominator to cancel common terms.
3. Substitute again after cancelling.

Worked Example

Find lim(x→1) of (x²−1)/(x−1).

Direct substitution gives 0/0. Factor: (x−1)(x+1)/(x−1) = x+1 for all x ≠ 1.

Now substitute: 1+1 = 2

Frequently Asked Questions

What if factoring doesn't work?

Try multiplying by a conjugate (for square roots) or L'Hopital's Rule (differentiating top and bottom separately) as alternative techniques for 0/0 forms.

What does it mean if left and right limits disagree?

The two-sided limit does not exist at that point — this commonly happens at jumps or breaks in a function, such as at x=0 for the function 1/x approaching from each side.