A reliable method
Try direct substitution for continuous expressions. If a quotient gives 0/0, simplify on a punctured neighborhood before substituting.
Worked foundation example
Find the limit as x approaches 3 of 2x² + 4. The polynomial is continuous at 3. Substitute x=3: 2(3)²+4. The limit is 22.
The common trap
Zero divided by zero is an indeterminate form, not the answer zero.
Connect the representations
For a two-sided limit, the left and right approaches must agree. At infinity, compare dominant powers rather than substituting infinity.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare a hole with a jump
For (x²−9)/(x−3), cancellation gives x+3 away from 3, so the limit is 6 even though the original expression is undefined there. In contrast, a function equal to 1 for x<3 and 2 for x>3 has incompatible one-sided limits. No single value assigned at 3 can repair that jump. Cancellation is a local algebraic argument, not permission to ignore all discontinuities.
Transfer and explain
Transfer: a function equals x+4 except at x=2, where its value is 99. Its limit at 2 is 6, not 99. Explain which part of the definition affects the limit and which part affects continuity.
Find the limit as x approaches 3 of 2x² + 4.
- The polynomial is continuous at 3.
- Substitute x=3: 2(3)²+4.
- The limit is 22.
Compare the other answer choices
Use b instead of b².
Discard the constant term.
Put the constant inside the squared input.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus