How to Use This Problem Set
Limits and continuity are foundational AP Calculus topics. They appear directly in multiple-choice questions and free-response problems, and they also provide the language needed for derivatives and later topics.
For each problem, work it out before opening the answer. Pay particular attention to whether the problem asks for a two-sided limit, a one-sided limit, a function value, or a continuity condition.
A function is continuous at when all three conditions hold: the function is defined at , the limit exists, and the limit equals the function value.
Problem Set
Evaluate .
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Evaluate .
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Evaluate .
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Evaluate .
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For the function , determine and .
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For the function , determine .
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Let . Determine the value of that makes continuous at .
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Determine whether is continuous at .
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Evaluate .
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Determine the behavior of as .
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Suppose is continuous on , with and . Explain why there must be at least one value such that .
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Let . Find the value of that makes continuous at .
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What to Check After You Finish
- Did you substitute first? Direct substitution should always be your first attempt when evaluating a limit.
- Did you recognize indeterminate forms? An expression such as does not mean the limit is zero or undefined. It tells you that additional algebra may be necessary.
- Did you compare one-sided limits? A two-sided limit exists only when the left-hand and right-hand limits agree.
- Did you distinguish a limit from a function value? The value of and the value of can be different.
- Did you check all three continuity conditions? A function must be defined, have a limit, and have that limit equal the function value.
Common Mistakes
The expression is an indeterminate form. It means that direct substitution has not finished the problem. Try factoring, rationalizing, using a known trigonometric limit, or another appropriate technique.
For to exist, the left-hand and right-hand limits must be equal. If they approach different values, the two-sided limit does not exist.
A function being defined at is necessary but not sufficient for continuity. You must also have .
When a denominator approaches zero, determine whether the expression approaches positive infinity or negative infinity from each side. The sign can change between the left and right sides.
Key Takeaways
Most AP Calculus limit problems can be organized into a small number of strategies:
- Substitute directly when the function is defined and produces a meaningful value.
- Factor and cancel when an algebraic expression produces .
- Rationalize when radicals produce an indeterminate form.
- Use one-sided limits when the behavior on either side of a point may differ.
- Check the three continuity conditions whenever a question asks whether a function is continuous.
- Use the Intermediate Value Theorem when continuity guarantees that a function must attain a value between two known endpoint values.
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