Key takeaway: A one-sided limit is the value a function approaches as x nears a number from only the left side or only the right side, written x→a⁻ or x→a⁺.
Grade 12 – AP Calculus ABAP Calculus AB Unit 1Calculus
Definition of One-Sided Limit
A one-sided limit looks at a function from a single direction. Walking toward \(a\) along the number line from smaller values gives the left-hand result; walking in from larger values gives the right-hand result.
Two directions can disagree, as happens at a jump in a graph or at a point where a formula changes. The ordinary two-sided value exists only when both sides head to the same number. Each one-sided limit is also the right tool for functions defined only on one side of a point, such as a square root at the edge of its domain.
Formula and notation
A minus sign in the superscript means from the left; a plus sign means from the right:
\[\lim_{x\to a^-}f(x)=L_1,\qquad \lim_{x\to a^+}f(x)=L_2.\]
The two-sided value exists exactly when \(L_1=L_2\), and then it equals that common number.
Key facts
- The superscript sign tells the side of approach, not the sign of the input.
- Disagreeing sides signal a jump; the two-sided value then does not exist.
- An infinite result from one side points to a vertical asymptote there.
- In a piecewise function, use the formula that covers the side you approach from.
One-Sided Limit example
A function follows the rule \(f(x)=3x-1\) for inputs below 2, and the rule \(f(x)=x^2+3\) for inputs of 2 or more. Find each one-sided value at \(x=2\), and decide whether the two-sided value exists.
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From the left, inputs are below 2, so use \(3x-1\): at 2 it gives \(3(2)-1=5\).
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From the right, use \(x^2+3\): at 2 it gives \(4+3=7\).
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The two results differ, \(5\neq 7\).
The left value is 5, the right value is 7, so the two-sided value does not exist.
Common mistake
Watch out
Zane finds \(f(2)=7\) from the second rule and says the answer from both sides must be 7.
Why it is wrong
The function value at the point can differ from what the left side approaches. Approaching from below 2 uses the first rule and heads for 5, so the sides disagree even though \(f(2)\) is defined.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus