One-Sided Limit
In short: A one-sided limit tracks the output a function heads toward when x closes in on a number while staying strictly on its left (x → c⁻) or strictly on its right (x → c⁺).
AP Calculus ABCalculusAP Calculus AB Unit 1
What is One-Sided Limit?
A one-sided limit watches a function while \(x\) approaches \(c\) in only one way. Along the number line you may arrive from the left (\(x < c\)) or from the right (\(x > c\)), and the two trips can end at different heights.
The notation is \(\lim_{x \to c^-} f(x)\) for the left and \(\lim_{x \to c^+} f(x)\) for the right. The two-sided value exists exactly when both of these exist and are equal. If they disagree, the graph jumps, and the overall value does not exist.
How to spot it
- The function uses different formulas on each side of a point, as in a tax schedule, so a one-sided limit is needed on each.
- A superscript plus or minus appears on the number the arrow points toward, as in x → 3⁺.
- A graph has a jump, with a hollow dot on one piece and a solid dot on another.
- An expression has an absolute value or a denominator that changes sign at the target point.
The method
- Pick the formula that applies to the side named: values less than \(c\) for the left, greater than \(c\) for the right.
- Plug \(c\) into that formula; when 0 over 0 shows up, simplify before plugging in.
- Do the same for the other side.
- Compare the two answers: equal numbers mean the two-sided value exists, unequal numbers mean it does not.
One-Sided Limit: solved example
Problem
A function is \(f(x) = 2x + 1\) for \(x < 3\) and \(f(x) = x^2 - 4\) for \(x \ge 3\). Find both one-sided values at \(x = 3\) and decide whether the two-sided value exists.
Solution
From the left, use \(2x + 1\): \(2(3) + 1 = 7\). From the right, use \(x^2 – 4\): \(9 – 4 = 5\).
Since \(7 \neq 5\), the two sides disagree. The left value is 7, the right value is 5, and the two-sided value does not exist, although \(f(3) = 5\).
Try it
Question 1
Find \(\lim_{x \to 0^+}\frac{1}{x}\) and \(\lim_{x \to 0^-}\frac{1}{x}\).
Show the answer
From the right the outputs grow without bound (\(+\infty\)); from the left they fall without bound (\(-\infty\)).
Question 2
Find both one-sided values of \(\frac{|x + 2|}{x + 2}\) at \(x = -2\).
Show the answer
For \(x > -2\) the quotient is 1; for \(x < -2\) it is -1. So right = 1, left = -1.
One-Sided Limit vs. Limit
The ordinary version needs the outputs to head to a single number from both directions at once. These halves are the pieces it is built from. If either half fails to exist, or the two halves differ, the two-sided number fails too, so checking both halves is the standard test.
Keep going
Lessons
Worksheets and quizzes
- Introducing Limits – Worksheet
- Introducing Limits – Quiz
- Continuity and the Intermediate Value Theorem – Worksheet
- Continuity and the Intermediate Value Theorem – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus