Key takeaway: A limit is the single height that a graph aims for while x creeps toward a target, regardless of whether the graph actually reaches that height.
Grade 12 – AP Calculus ABAP Calculus AB Unit 1Calculus
Definition of Limit
A limit describes where a function is heading, not where it is. Pick an input value \(a\), move \(x\) toward it from both sides, and watch the outputs. If they settle toward a single number \(L\), then \(L\) is the answer.
The function does not have to be defined at \(a\); it can even have a hole there. That freedom makes the idea the foundation of calculus, since derivatives and integrals are both built from limits. Three tools estimate one: a table of values near \(a\), a graph, and algebra such as factoring to remove a zero from the denominator.
Formula and notation
Read this as “the limit of f(x) as x approaches a equals L”:
\[\lim_{x\to a} f(x)=L.\]
It means \(f(x)\) can be made as close to \(L\) as you like by taking \(x\) close enough to \(a\), with \(x\neq a\). The value \(f(a)\) itself plays no role.
Key facts
- The outputs can approach a number even when f(a) is undefined or different.
- Direct substitution works whenever the function is continuous at a.
- A 0/0 result from substitution is a signal to factor or simplify, not to stop.
- If the left-hand and right-hand outputs disagree, the two-sided value does not exist.
Limit example
Find \(\lim_{x\to -1}\dfrac{x^3+1}{x+1}\).
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Substituting \(-1\) gives \(\frac{0}{0}\), which has no meaning, so simplify first.
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The top is a sum of cubes: \(x^3+1=(x+1)(x^2-x+1)\). Cancel the shared factor \(x+1\) for every \(x\neq -1\).
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What remains is \(x^2-x+1\); substitute \(-1\): \(1+1+1=3\).
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Table check: at \(x=-1.01\) the quotient is 3.0301, and at \(x=-0.99\) it is 2.9701.
The value is 3, even though the function itself is undefined at \(x=-1\).
Common mistake
Watch out
Ines substitutes \(x=-1\), sees \(\frac{0}{0}\), and writes that the answer does not exist.
Why it is wrong
The form \(\frac{0}{0}\) is indeterminate: it says nothing yet. After canceling the common factor, the outputs head for 3, so the answer exists.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus