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Teacher Guide: One-Sided Limits

Learn to find left-hand and right-hand limits from graphs and piecewise formulas, and decide when a two-sided limit exists.

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10 questions on Limits. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Turn x→a−x \to a^- and x→a+x \to a^+ into concrete xx-values on the correct side of aa (like 2.992.99 and 3.013.01 for a=3a = 3)
  • Find left-hand and right-hand limits from a graph with open circles and closed dots
  • Evaluate one-sided limits of piecewise functions and expressions with absolute values
  • Decide whether a two-sided limit exists by comparing the one-sided limits
  • Find a constant that makes the one-sided limits of a piecewise function equal
Prerequisites
  • • Introduction to Limits and Limit Notation (the meaning of lim⁡x→af(x)\lim_{x \to a} f(x))
  • • Evaluating Limits Graphically and Evaluating Limits from Tables
  • • Limits by Direct Substitution
  • • Piecewise functions and absolute value
Discussion Starters
  • 1. Why can a function have a value at x=ax = a but no limit there?
  • 2. A parking fee jumps at every full hour. What do the left-hand and right-hand limits tell a driver?
  • 3. Can you draw a graph where both one-sided limits at x=2x = 2 equal 33, but f(2)=5f(2) = 5?
  • 4. Why do we need the word "and" in the rule: the limit exists if both one-sided limits exist and are equal?
Common Misconceptions

The limit is the yy-value of the filled dot

Remediation: Show Example 3: the graph approaches 44 from both sides while g(2)=7g(2) = 7. Cover the point x=ax = a with a finger and ask what height each side is heading to.

a−a^- means a negative number

Remediation: Write 3−→2.9,2.99,2.9993^- \to 2.9, 2.99, 2.999 and 3+→3.1,3.01,3.0013^+ \to 3.1, 3.01, 3.001 on the board. Point out that 3−3^- is still close to +3+3.

If both one-sided limits exist, the limit exists

Remediation: Use ∣x∣x\frac{|x|}{x}: both one-sided limits exist (−1-1 and 11), yet there is no single value. Stress that the two numbers must also be equal.

Differentiation Ideas

For Struggling Students:

  • • Start with step graphs where each piece is horizontal, so each one-sided limit can be read directly
  • • Use tables with x=0.9,0.99,0.999x = 0.9, 0.99, 0.999 and 1.1,1.01,1.0011.1, 1.01, 1.001 next to the graph
  • • Color the left piece and the right piece in different colors

For On-Level Students:

  • • Evaluate one-sided limits of piecewise functions given only by formulas
  • • Simplify ∣x−a∣x−a\frac{|x - a|}{x - a} and similar expressions on each side
  • • Explain in words why a limit does or does not exist

For Advanced Students:

  • • Find constants that make a piecewise function have a limit at a breakpoint
  • • Investigate functions with three pieces and two breakpoints
  • • Preview: compare lim⁡x→0−\lim_{x \to 0^-} and lim⁡x→0+\lim_{x \to 0^+} of ∣x∣x\frac{|x|}{x} with the slopes of ∣x∣|x| on each side
Standards Alignment
  • AP Calculus LIM-1.A (AP.CALC.LIM-1.A)

    Represent limits analytically using correct notation, including one-sided limits

  • AP Calculus LIM-1.C (AP.CALC.LIM-1.C)

    Estimate limits of functions from graphs and tables, including one-sided limits

  • HSF-IF.C.7b (CCSS.MATH.CONTENT.HSF.IF.C.7.B)

    Graph piecewise-defined functions, including step functions and absolute value functions

Lesson Resources
  • visualJump Slider Graph

    Move the slider bb to shift the right piece; the limits panel shows when both sides meet

  • activityLeft or Right?

    Students sort cards with xx values such as 2.992.99 and 3.013.01 into the groups x→3−x \to 3^- and x→3+x \to 3^+

  • worksheetPiecewise Limits Practice

    Find left, right and two-sided limits of piecewise and absolute value functions

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A one-sided limit looks at a function from only one side of a point.
  • The left-hand limit lim⁡x→a−f(x)=L\lim_{x \to a^-} f(x) = L means: as xx gets close to aa through values less than aa (like 1.9,1.99,1.9991.9, 1.99, 1.999 for a=2a = 2), f(x)f(x) gets close to LL.
  • The right-hand limit lim⁡x→a+f(x)=M\lim_{x \to a^+} f(x) = M means: as xx gets close to aa through values greater than aa (like 2.1,2.01,2.0012.1, 2.01, 2.001), f(x)f(x) gets close to MM.
You know this notation from the lesson Limit Notation: the small sign shows the side and is not part of the number, so 3−3^- is not −3-3. In this lesson you compute these limits and use them to decide whether lim⁡x→af(x)\lim_{x \to a} f(x) exists.
The key rule connects one-sided and two-sided limits:
lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=L and lim⁡x→a+f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L
If the two one-sided limits are different, the two-sided limit does not exist (DNE). The value f(a)f(a) itself never matters for any of these limits: the function may be undefined at aa, or have a completely different value there.

Worked Examples

The graph of ff shows the line y=x+1y = x + 1 for x<1x < 1 (open circle at (1,2)(1, 2)) and the line y=x+3y = x + 3 for x≥1x \ge 1 (closed dot at (1,4)(1, 4)). Find lim⁡x→1−f(x)\lim_{x \to 1^-} f(x), lim⁡x→1+f(x)\lim_{x \to 1^+} f(x) and lim⁡x→1f(x)\lim_{x \to 1} f(x).
1

Follow the graph from the left

Trace the left part toward x=1x = 1: the heights approach the open circle at y=2y = 2 → lim⁡x→1−f(x)=2\lim_{x \to 1^-} f(x) = 2
2

Follow the graph from the right

Trace the right part toward x=1x = 1: the heights approach y=4y = 4 → lim⁡x→1+f(x)=4\lim_{x \to 1^+} f(x) = 4
3

Compare the two sides

2≠42 \neq 4, so the graph jumps at x=1x = 1 → The one-sided limits differ
4

Decide about the two-sided limit

Different one-sided limits mean no single value is approached. The closed dot f(1)=4f(1) = 4 does not change this → lim⁡x→1f(x)\lim_{x \to 1} f(x) does not exist

Common Mistakes

Using the closed dot f(a)f(a) as the limit

Why it's wrong: A limit describes the heights the graph approaches near aa, not the height at aa. In Example 1, f(1)=4f(1) = 4, but from the left the graph approaches 22.

Correct: Trace each side of the graph toward aa and ignore the dot at x=ax = a. Use f(a)f(a) only when the question asks for the value.

Reading x→3−x \to 3^- as "xx approaches −3-3"

Why it's wrong: The small minus sign is written after the number, as a superscript. It describes the direction, not the sign of the number.

Correct: x→3−x \to 3^- means x=2.9,2.99,2.999,…x = 2.9, 2.99, 2.999, \dots (values below 33). x→3+x \to 3^+ means x=3.1,3.01,3.001,…x = 3.1, 3.01, 3.001, \dots

Using the piece that contains x=ax = a for both sides

Why it's wrong: In f(x)=x+1f(x) = x + 1 for x<1x < 1 and f(x)=x+3f(x) = x + 3 for x≥1x \ge 1, the piece x+3x + 3 only describes the right side. The left side is always given by the piece for x<1x < 1.

Correct: For x→a−x \to a^- use the formula for values just below aa; for x→a+x \to a^+ use the formula for values just above aa. The ≤\le or ≥\ge sign only decides f(a)f(a).

Saying the limit exists because both one-sided limits exist

Why it's wrong: Both one-sided limits in Example 1 exist (22 and 44), but they are different numbers, so there is no single value LL.

Correct: The two-sided limit exists only if both one-sided limits exist and are equal.

Why It Matters

One-sided limits are the tool for functions that change their rule at a point:
  • Piecewise functions: tax brackets, phone plans and shipping prices switch formulas at certain values, and you must look at each side separately
  • Jumps: a parking fee that jumps from 3 to 5 dollars after one hour has different left-hand and right-hand limits
  • Absolute values: expressions like ∣x∣x\frac{|x|}{x} behave differently for negative and positive xx
  • Continuity: in the lessons Continuity and Types of Discontinuity you will check one-sided limits to classify every break in a graph
  • Calculus: derivatives of functions like ∣x∣|x| are studied with left and right limits
Whenever a graph has a break, the first question is always: what happens on the left, and what happens on the right?

Real World Applications

Parking Fees

Many parking garages charge by started hour, so the price jumps at each full hour. One-sided limits describe the price just before and just after the jump.

Example:

A garage charges P(t)=3P(t) = 3 dollars for 0<t≤10 < t \le 1 hour and P(t)=5P(t) = 5 dollars for 1<t≤21 < t \le 2 hours. Just before one hour you pay 3 dollars, just after it 5 dollars.

1Try It Yourself

The garage uses P(t)=3P(t) = 3 for 0<t≤10 < t \le 1 and P(t)=5P(t) = 5 for 1<t≤21 < t \le 2 (tt in hours, price in dollars).

By how many dollars does the price jump at t=1t = 1?

Step 1: Write the mathematical expression

Right-hand limit minus left-hand limit at t=1t = 1:

Building a Smooth Ramp

Engineers who join a ramp to a platform need the heights from both sides to meet. Otherwise there is a step that people can trip over.

Example:

A ramp has height h(x)=0.2xh(x) = 0.2x meters for 0≤x<50 \le x < 5, and the platform starts at x=5x = 5 at a constant height cc. The ramp and the platform meet without a step exactly when lim⁡x→5−h(x)=c\lim_{x \to 5^-} h(x) = c.

2Try It Yourself

The ramp has height h(x)=0.2xh(x) = 0.2x for 0≤x<50 \le x < 5 and the platform has height cc for x≥5x \ge 5 (all lengths in meters).

Which platform height cc gives no step at x=5x = 5?

Step 1: Write the mathematical expression

The platform height must equal the left-hand limit at x=5x = 5:

Mobile Data Plans

A data plan often has a fixed price up to a limit and an extra charge per gigabyte after it. One-sided limits show whether the price jumps at the limit.

Example:

A plan costs C(g)=10C(g) = 10 dollars for 0≤g≤50 \le g \le 5 GB and C(g)=10+2(g−5)C(g) = 10 + 2(g - 5) dollars for g>5g > 5. Just above 5 GB the price starts to grow, but it starts from 10.

3Try It Yourself

The plan costs C(g)=10C(g) = 10 for 0≤g≤50 \le g \le 5 and C(g)=10+2(g−5)C(g) = 10 + 2(g - 5) for g>5g > 5 (gg in GB, price in dollars).

What is lim⁡g→5C(g)\lim_{g \to 5} C(g)?

Step 1: Write the mathematical expression

Find both one-sided limits at g=5g = 5 and compare them:

Key Takeaways

  • 1lim⁡x→a−f(x)\lim_{x \to a^-} f(x) is the left-hand limit: xx approaches aa through values less than aa
  • 2lim⁡x→a+f(x)\lim_{x \to a^+} f(x) is the right-hand limit: xx approaches aa through values greater than aa
  • 3lim⁡x→af(x)=L\lim_{x \to a} f(x) = L exactly when both one-sided limits exist and both equal LL
  • 4If the one-sided limits are different (a jump), the two-sided limit does not exist
  • 5For a piecewise function, use the piece for values just below aa on the left and the piece for values just above aa on the right
  • 6The value f(a)f(a) (open circle or closed dot) never changes a limit

Frequently Asked Questions

Can a one-sided limit exist when the two-sided limit does not?

Yes, and this is the typical case at a jump. For ∣x∣x\frac{|x|}{x} at 00 both one-sided limits exist (−1-1 and 11), but the two-sided limit does not, because they are different.

Does the closed dot or the open circle decide the limit?

Neither. Circles and dots only show the value f(a)f(a). The limits come from the curve on each side as it approaches x=ax = a.

What if the function is only defined on one side?

Then only one one-sided limit makes sense. For x\sqrt{x} at 00 there are no values to the left, so we use lim⁡x→0+x=0\lim_{x \to 0^+} \sqrt{x} = 0. At the end of a domain, the one-sided limit is the natural question.

Can a one-sided limit be infinite?

Yes. For example 1x\frac{1}{x} grows without bound as x→0+x \to 0^+. These cases are covered in the lesson Infinite Limits; in this lesson all one-sided limits are numbers.

Glossary

One-sided limit
A limit that uses values of xx from only one side of aa
Left-hand limit
The value f(x)f(x) approaches as x→a−x \to a^-, through values less than aa
Right-hand limit
The value f(x)f(x) approaches as x→a+x \to a^+, through values greater than aa
Two-sided limit
The ordinary limit lim⁡x→af(x)\lim_{x \to a} f(x); it exists only if the left-hand and right-hand limits are equal
Piecewise function
A function given by different formulas on different intervals of xx
Jump discontinuity
A break where both one-sided limits exist but are different, so the graph jumps

Formula Card

Left-hand limit

lim⁡x→a−f(x)\lim_{x \to a^-} f(x)

$x$ approaches $a$ from the left (values less than $a$)

Right-hand limit

lim⁡x→a+f(x)\lim_{x \to a^+} f(x)

$x$ approaches $a$ from the right (values greater than $a$)

When the limit exists

lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

Both one-sided limits must exist and be equal

Absolute value by sides

∣x−a∣=x−a|x - a| = x - a for x>ax > a, ∣x−a∣=−(x−a)|x - a| = -(x - a) for x<ax < a

Use this to simplify expressions with absolute values on each side

More in This Topic

Introduction to Limits

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Limit Notation

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Evaluating Limits Graphically

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Evaluating Limits from Tables

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