Evaluate limits numerically, graphically, and algebraically, including limits at infinity, and describe what it means for a function to be continuous at a point.
A limit describes what a function's output APPROACHES as the input gets closer and closer to some value -- not necessarily what the function actually equals there. Notation: lim as x approaches a of f(x) = L means f(x) gets arbitrarily close to L as x gets close to a, from either side. A limit can exist perfectly well even at a value where the function itself is undefined, which is exactly what happens at a hole.
When direct substitution gives a real number, that number IS the limit. When it gives the indeterminate form 0/0, factor and cancel first (exactly like finding a rational function's hole), then substitute again. Limits AT infinity describe end behavior: for a rational function, they follow the exact same degree-comparison rules from the horizontal-asymptote unit -- the limit as x approaches infinity of a rational function equals its horizontal asymptote, when one exists. A function is continuous at x=a when the limit as x approaches a exists, f(a) is defined, and the two are EQUAL -- all three conditions, with no exceptions.
Mastery looks like: They evaluate limits numerically, graphically, and algebraically (including at infinity), and correctly determine whether a function is continuous at a given point.
Common stumbles: Treating 0/0 as proof that a limit does not exist instead of a signal to simplify; concluding a function is continuous just because a limit exists, without checking that it matches the function's actual value there.