AP Calculus AB · Lesson 1 of 5
Functions, limits, and continuity
Separate a function value from a limit, and state the conditions that make a function continuous at a point.
Limits
The limit of f(x) as x approaches a describes the values f approaches, not necessarily f(a). A hole in a graph can make the limit exist while the function value is undefined or different. One-sided limits must agree for the two-sided limit to exist.
If direct substitution gives a number for a polynomial or a rational function whose denominator is not zero, that number is the limit. If substitution gives 0/0, the expression may have a common factor, and canceling that factor can reveal the limit. The canceled value may still not be in the domain.
Continuity
A function is continuous at a if it is defined there, the limit exists there, and the limit equals the function value. Continuity on a closed interval matters because several theorems, including the intermediate value theorem, require it.
A jump, a hole, or a vertical asymptote is a reason continuity fails. Pointing at a graph is not a justification unless you connect it to one of those three conditions.
Worked example
Let f(x) = (x² − 4)/(x − 2) for x ≠ 2, and suppose f(2) is not defined. What is the limit as x approaches 2?
- Factor: (x − 2)(x + 2)/(x − 2).
- For x ≠ 2, f(x) = x + 2.
- As x approaches 2, the expression approaches 4, even though f(2) is not defined.
Why this works. The limit can exist at a point where the function does not.
Check your understanding
- State whether you are finding a function value or a limit.
- Check both one-sided limits when a piecewise definition changes.
- Name which condition of continuity fails if the function is discontinuous.