Limits and Continuity

Calculus 3 · Axiom Academy

LESSON Limits and Continuity for Vector Functions Extending single-variable limit and continuity concepts to vector-valued functions through componentwise analysis This means we can find limits of vector functions by finding limits of each component separately. If any component limit fails to exist, the vector limit doesn't exist. 2. Continuity of Vector Functions Geometrically, continuity means the curve traced by (t) has no breaks, jumps, or holes at t = a . A jump discontinuity occurs when the one-sided limits exist but are not equal. Consider: At t = 0 , the x -component jumps from -1 to 1 , creating a discontinuity in the curve. The left and right limits exist but differ: A removable discontinuity occurs when the limit exists but either the function is undefined at the point or has the wrong value. Consider: At t = 0 , the function is undefined (a 0/0 form in the first component), but the limit exists: We can make continuous by defining (0) = 1, 0 , "removing" the discontinuity. The one-sided limits both exist but disagree, so _ t a (t) does not exist at all. No single value of (a) can repair it. The limit does exist; only the value at t = a is missing or wrong. Set (a) equal to that limit and the curve closes up. Limits and continuity for a vector function are never new machinery — they are the single-variable ideas you already know, run once per component. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Calculus 3 course.