Skip to main content
Mathovia

Calculus / Calculus 1 · Limits

One-Sided Limits

A one-sided limit only looks at one side of x = a: the left-hand limit uses values just below a, and the right-hand limit uses values just above it. They're the tool for jumps, endpoints and asymptotes, and the two-sided limit exists exactly when both of them agree.

Practice Problems
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Chapter 2 · Lesson 3 of 13Calculus 1 · Limits
  1. 1.Introduction to Limits
  2. 2.Finding Limits From a Graph
  3. 3.One-Sided Limits
  4. 4.Limit Laws
  5. 5.Evaluating Limits
  6. 6.Squeeze Theorem
  7. 7.Infinite Limits
  8. 8.Limits at Infinity
  9. 9.Continuity
  10. 10.Types of Discontinuity
  11. 11.Intermediate Value Theorem
  12. 12.Epsilon-Delta Definition of a Limit
  13. 13.L'Hôpital's Rule
See all calculus chapters →

Some functions behave differently on each side of a point. A piecewise function can switch formulas, a square root can stop at the edge of its domain, and a fraction can blow up in opposite directions. In each case it helps to ask about one side at a time. That’s what one-sided limits do, and they’re how you decide whether an ordinary two-sided limit exists at all.

Notation: the limit of f of x as x approaches a from the left, written x to a minus, and the limit as x approaches a from the right, written x to a plus
Left-hand and right-hand limits

What Is a One-Sided Limit?

The left-hand limit is the value approaches as moves towards through values smaller than . The right-hand limit uses values larger than .

A number line with a marked point a. Orange dots approach a from the left, labelled x to a minus, the left-hand limit, x less than a. Violet dots approach from the right, labelled x to a plus, the right-hand limit, x greater than a
The sign shows the direction of approach

The small minus and plus are directions only. means runs through , and means . The two kinds of limit connect to the ordinary limit by one rule:

That rule is what you were using in finding limits from a graph when you traced the curve from each side.

How to Find One-Sided Limits

  1. Decide which values of count: just below for , just above for .
  2. Use only the rule that applies there. For a piecewise function, that means one piece.
  3. Substitute, simplify or check signs, exactly as for any limit.
  4. Compare the two sides if the question asks about the two-sided limit.

Example 1: A Piecewise Function With a Jump

Find the one-sided limits at for

From the left, , so use : .

From the right, , so use : .

Graph of x squared for x less than 1 ending in an open circle at (1, 1), and the line 3x minus 1 for x at least 1 starting with a filled dot at (1, 2). The left-hand limit is 1 and the right-hand limit is 2
Left-hand limit 1, right-hand limit 2

The sides disagree, so does not exist. The function does have a value, though: , from the piece that includes .

Example 2: Pieces That Meet

Find for when and when .

From the left: . From the right: . Both sides give 5, so . The pieces join up with no jump, which is why checking both sides matters even when you expect a break.

Example 3: An Absolute Value Quotient

Find the one-sided limits of at .

Remember that when and when (see absolute value functions).

  • For , is positive, so the fraction is . The right-hand limit is 1.
  • For , is negative, so the fraction is . The left-hand limit is .

The two-sided limit does not exist. Fractions with an absolute value over the same expression always behave like this: they’re a jump from to 1.

Example 4: The Edge of a Domain

Find .

As shrinks towards 0 through positive values, shrinks towards 0, so the right-hand limit is 0. There’s no left-hand limit, because isn’t defined for . At the endpoint of a domain, the one-sided limit from inside the domain is the only one that makes sense. The square root functions lesson shows the graph starting at the origin.

Example 5: Infinite One-Sided Limits

Describe as and .

The numerator stays at 1 while the denominator shrinks to 0, so the size of the fraction grows without bound. The sign decides the direction:

  • Just right of 2, e.g. , the denominator is and the fraction is . So .
  • Just left of 2, e.g. , the denominator is and the fraction is . So .
Graph of 1 over (x minus 2) with a dashed vertical asymptote at x = 2. To the left the curve falls to negative infinity; to the right it rises to positive infinity
Opposite infinities on the two sides of x = 2

This sign check is the quickest way to describe a function near a vertical asymptote. The lesson on infinite limits builds on it.

Example 6: A Step Function

The floor function rounds down to the nearest integer, so and . Find its one-sided limits at .

Staircase graph of the floor function from 0 to 5. Each step has a filled dot on the left and an open circle on the right. At x = 3 the step at height 2 ends with an open circle and the step at height 3 begins with a filled dot
The floor function jumps at every integer

Just below 3, values like 2.99 round down to 2, so . Just above 3, values like 3.01 round down to 3, so . The two-sided limit fails at every integer, but between integers, say at , both sides give 2 and the limit exists.

Step functions like this model real prices. A parking garage that charges 4 dollars for each started hour, or postage that rises at each weight band, has a jump at every boundary, and the one-sided limits tell you the price just before and just after it.

Common Mistakes to Avoid

  • Reading the minus sign as a negative number. approaches positive 2 from below.
  • Using the wrong piece. For , use the piece for , even if the piece for includes itself.
  • Assuming the two-sided limit exists because one side does. Both sides must exist and agree.
  • Guessing the sign of an infinite limit. Test a value just beside to see whether the function heads up or down.

Practice Problems

Work each one before opening the answer.

Problem 1. For when and when , find and .

Show answer

Left: . Right: .

Answer: 5 from the left, 2 from the right, so the two-sided limit does not exist

Problem 2. Find .

Show answer

For , , so the fraction is .

Answer:

Problem 3. Find .

Show answer

For just above 4, is a small positive number and its square root approaches 0.

Answer: 0

Problem 4. Describe .

Show answer

At the denominator is and the fraction is . The values fall without bound.

Answer:

Problem 5. Find and .

Show answer

Just below 1, values round down to 0; just above 1, they round down to 1.

Answer: 0 from the left, 1 from the right

Next come the limit laws: the rules that let you split a limit into sums, products and quotients and work it out without a table or a graph.

Frequently Asked Questions

What is a one-sided limit?+

A one-sided limit is the value approaches as gets close to from one side only. The left-hand limit uses ; the right-hand limit uses .

What do the minus and plus signs in x → a⁻ and x → a⁺ mean?+

They show the direction of approach, not the sign of a number. means approaches 3 through values like 2.9, 2.99, 2.999; means through 3.1, 3.01, 3.001.

How do you find one-sided limits?+

Use only the part of the function that applies on that side. For a piecewise function, pick the piece for or and substitute into it. For expressions like or , work out the sign of each part for just below or just above .

When does a two-sided limit exist?+

exactly when both one-sided limits exist and both equal . If either side fails, or the two sides differ, the two-sided limit does not exist.

Can a one-sided limit be infinite?+

Yes, in the sense of describing behaviour: says the values grow without bound as approaches 2 from the right. Strictly, the limit still doesn't exist as a number.

Related lessons

Finding Limits From a GraphIntroduction to LimitsPiecewise Functions: Different Rules on Different IntervalsAbsolute Value Functions: The V-Shaped Graph