Limits, Continuity, and Limit Theorems
This topic tests limits from graphs, tables, and formulas; one-sided limits; infinite limits; continuity; asymptotes; squeeze theorem; and intermediate value theorem.
How to study for AP Calculus BC
Build every answer from the representation first: identify the quantity, choose the theorem or process, show notation, and justify the conclusion with conditions.
Core concepts
Concept 1
Limit questions require checking the approach behavior, not only the function value.
Exam cue: Compare left and right behavior before claiming a two-sided limit.
Concept 2
Continuity needs three conditions at a point.
Exam cue: Check whether the function value is defined and equals the limit.
Concept 3
Theorems require hypotheses such as continuity on an interval.
Exam cue: State continuity hypotheses when using IVT.
Risk pitfalls and guardrails
Using the value at the point as the limit without checking nearby behavior.
Guardrail: Avoid answers that rely only on habit, ignore the stated source, skip safety or compliance steps, or choose convenience over the professional standard.
Applying IVT to a function that is not continuous on the interval.
Guardrail: Avoid answers that rely only on habit, ignore the stated source, skip safety or compliance steps, or choose convenience over the professional standard.
Confusing vertical asymptotes with removable discontinuities.
Guardrail: Avoid answers that rely only on habit, ignore the stated source, skip safety or compliance steps, or choose convenience over the professional standard.
Memory anchors
Limit
A limit describes the value a function approaches as the input approaches a point.
One-Sided Limit
A one-sided limit approaches from only the left or only the right.
Continuity
Continuity at a point requires the function value, limit, and equality between them.
Removable Discontinuity
A removable discontinuity can be repaired by redefining one function value.
Infinite Limit
An infinite limit describes unbounded behavior near an input.
Squeeze Theorem
The squeeze theorem finds a limit by trapping a function between two functions with the same limit.
Intermediate Value Theorem
The intermediate value theorem guarantees values on an interval for continuous functions.
Asymptote
An asymptote describes limiting behavior of a graph.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Evaluate lim as x approaches 3 of (x^2 - 9)/(x - 3).
Evaluate lim as x approaches 0 of sin(x)/x.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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