Limit
In short: A limit names the output that a function steers toward as its input creeps up on a target number, which can differ from the output the function actually has there.
AP Calculus ABCalculusAP Calculus AB Unit 1
What is Limit?
A limit records the destination of a function’s outputs rather than the output at one exact input. Slide the input toward a number \(c\) from both sides, and the outputs may close in on a single value \(L\), even when the graph has a hole at \(c\) or no value there at all.
The statement \(\lim_{x \to c} f(x) = L\) promises that \(f(x)\) stays within any tolerance of \(L\), however tiny, once \(x\) is near enough to \(c\) (but not equal to it). Such a number exists only when the left-hand and right-hand approaches agree. The idea underlies derivatives and integrals.
How to spot it
- In a table, the outputs steady near one number while the inputs close in on a target value.
- A graph has an open circle (a hole) and the question asks what height the curve is aiming for.
- A formula gives 0/0 when you substitute, which signals that algebra is needed before the answer appears.
- The notation lim with an arrow under it, x → c, appears in front of an expression.
The method
- Plug \(c\) straight into the formula; a finite number out means you are done.
- If you get \(\frac{0}{0}\), factor the numerator and denominator and cancel the common factor.
- Substitute \(c\) into the simplified expression.
- Check with a table of values or a graph that both sides approach the same number.
Limit: solved example
Problem
Evaluate \(\lim_{x \to -4}\dfrac{x^2 + 3x – 4}{x + 4}\).
Solution
Direct substitution gives \(\frac{16 – 12 – 4}{0} = \frac{0}{0}\), so factor: \(x^2 + 3x – 4 = (x + 4)(x – 1)\).
For \(x \neq -4\) the fraction equals \(x – 1\). Substituting \(-4\) gives \(-5\). The value is \(-5\), even though the original expression is undefined at \(x = -4\).
Try it
Question 1
Evaluate \(\lim_{x \to 4}(3x – 5)\).
Show the answer
Substitution is safe: \(3(4) – 5 = 7\).
Question 2
Evaluate \(\lim_{x \to -1}\dfrac{x^2 + x}{x + 1}\).
Show the answer
Factor the top as \(x(x + 1)\), cancel, and substitute: \(-1\).
Limit vs. Continuity
A limit asks only where the outputs are heading as the input nears a spot, and ignores what happens exactly there. Continuity demands more: the function must be defined at the spot, the approach value must exist, and the two must be equal. A graph with a hole has a value heading somewhere but is not continuous there.
Keep going
Lessons
Worksheets and quizzes
- Introducing Limits – Worksheet
- Introducing Limits – Quiz
- Evaluating Limits Algebraically – Worksheet
- Evaluating Limits Algebraically – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus