Limits and Continuity

What a limit means, how to compute one, and when a function is continuous.

A limit describes the value f(x)f(x) approaches as xx gets arbitrarily close to some target aa — not the value at aa itself. The distinction matters whenever ff is undefined, jumps, or has a hole at aa.

How to evaluate a limit

  1. <strong>Try direct substitution.</strong> If f(a)f(a) is defined and continuous there, the limit is just f(a)f(a).
  2. <strong>If you get 0/00/0, factor or simplify.</strong> Cancel the common factor that's causing the indeterminate form, then substitute again.
  3. <strong>If algebra fails, use special techniques</strong> — conjugates for radicals, common denominators for compound fractions, or L'Hôpital's rule (after derivatives).
  4. <strong>For limits at infinity</strong>, compare degrees of the numerator and denominator, or factor out the dominant term.

Continuity, in one sentence

ff is continuous at aa iff lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a). Three failure modes: the limit doesn't exist (jump), f(a)f(a) is undefined (hole), or the two are unequal (removable discontinuity).