Limits and Continuity
Sixteen topics on what a function is doing as you close in on a point, and what it takes for the graph to be unbroken there. Rates at an instant and the secants that define them, limits read from graphs, tables and algebra, the Squeeze Theorem for functions with no limit of their own, the three kinds of discontinuity and which one can be repaired, continuity at a point and across an interval, vertical and horizontal asymptotes, and the Intermediate Value Theorem.
AB exam 10-15%BC exam 5-10%16 topics
Topics
Key forms For every problem in this unit
Average rate on [a, b]
(f(b) − f(a)) / (b − a)
Instantaneous rate at a
limit of that quotient as b → a
Two-sided limit exists
left limit = right limit (and both exist)
Indeterminate forms
0/0, ∞/∞, 0·∞, ∞−∞: do more work
Determinate: k/0, k ≠ 0
unbounded, so no limit
Squeeze Theorem
g ≤ f ≤ h near a, and g, h → L, so f → L
Standard trig limit
sin(u)/u → 1 as u → 0
Continuous at a
f(a) defined, limit exists, and the two are equal
Removable (hole)
sides agree, value missing or different
Jump
sides finite and unequal
Infinite
a side unbounded: vertical asymptote
Vertical asymptote
factor SURVIVES in the reduced denominator
Hole
factor CANCELS against the numerator
Horizontal asymptote
top < bottom → 0; equal → ratio of leading coefficients; top > bottom → none
IVT
f continuous on [a, b], N between f(a) and f(b): f(c) = N for some c