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.

Present LessonView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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

For Teachers

Learning Objectives

Turn $x \to a^-$ and $x \to a^+$ into concrete $x$ -values on the correct side of $a$ (like $2.99$ and $3.01$ for $a = 3$ )
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 $\lim_{x \to a} f(x)$ )
• 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 $x = a$ 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 $x = 2$ equal $3$ , but $f(2) = 5$ ?
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 $y$ -value of the filled dot

Remediation: Show Example 3: the graph approaches $4$ from both sides while $g(2) = 7$ . Cover the point $x = a$ with a finger and ask what height each side is heading to.

$a^-$ means a negative number

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

If both one-sided limits exist, the limit exists

Remediation: Use $\frac{|x|}{x}$ : both one-sided limits exist ( $-1$ and $1$ ), 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 $x = 0.9, 0.99, 0.999$ and $1.1, 1.01, 1.001$ 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 $\frac{|x - a|}{x - a}$ 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 $\lim_{x \to 0^-}$ and $\lim_{x \to 0^+}$ of $\frac{|x|}{x}$ with the slopes of $|x|$ 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 $b$ to shift the right piece; the limits panel shows when both sides meet
activityLeft or Right?
Students sort cards with $x$ values such as $2.99$ and $3.01$ into the groups $x \to 3^-$ and $x \to 3^+$
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 $\lim_{x \to a^-} f(x) = L$ means: as $x$ gets close to $a$ through values less than $a$ (like $1.9, 1.99, 1.999$ for $a = 2$ ), $f(x)$ gets close to $L$ .
The right-hand limit $\lim_{x \to a^+} f(x) = M$ means: as $x$ gets close to $a$ through values greater than $a$ (like $2.1, 2.01, 2.001$ ), $f(x)$ gets close to $M$ .

You know this notation from the lesson Limit Notation: the small sign shows the side and is not part of the number, so

3^-

is not

-3

. In this lesson you compute these limits and use them to decide whether

\lim_{x \to a} f(x)

exists.

The key rule connects one-sided and two-sided limits:

\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)

itself never matters for any of these limits: the function may be undefined at

a

, or have a completely different value there.

Worked Examples

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

Follow the graph from the left

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

Follow the graph from the right

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

Compare the two sides

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

Decide about the two-sided limit

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

Answer: $\lim_{x \to 1^-} f(x) = 2$ , $\lim_{x \to 1^+} f(x) = 4$ , so $\lim_{x \to 1} f(x)$ does not exist (even though $f(1) = 4$ ).

Common Mistakes

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

Why it's wrong: A limit describes the heights the graph approaches near $a$ , not the height at $a$ . In Example 1, $f(1) = 4$ , but from the left the graph approaches $2$ .

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

Reading $x \to 3^-$ as " $x$ approaches $-3$ "

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: $x \to 3^-$ means $x = 2.9, 2.99, 2.999, \dots$ (values below $3$ ). $x \to 3^+$ means $x = 3.1, 3.01, 3.001, \dots$

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

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

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

Saying the limit exists because both one-sided limits exist

Why it's wrong: Both one-sided limits in Example 1 exist ( $2$ and $4$ ), but they are different numbers, so there is no single value $L$ .

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 $\frac{|x|}{x}$ behave differently for negative and positive $x$
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 $|x|$ 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 $P(t) = 3$ dollars for $0 < t \le 1$ hour and $P(t) = 5$ dollars for $1 < t \le 2$ hours. Just before one hour you pay 3 dollars, just after it 5 dollars.

1Try It Yourself

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

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

Step 1: Write the mathematical expression

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

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 $h(x) = 0.2x$ meters for $0 \le x < 5$ , and the platform starts at $x = 5$ at a constant height $c$ . The ramp and the platform meet without a step exactly when $\lim_{x \to 5^-} h(x) = c$ .

2Try It Yourself

The ramp has height $h(x) = 0.2x$ for $0 \le x < 5$ and the platform has height $c$ for $x \ge 5$ (all lengths in meters).

Which platform height $c$ gives no step at $x = 5$ ?

Step 1: Write the mathematical expression

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

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

3Try It Yourself

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

What is $\lim_{g \to 5} C(g)$ ?

Step 1: Write the mathematical expression

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

Key Takeaways

1 $\lim_{x \to a^-} f(x)$ is the left-hand limit: $x$ approaches $a$ through values less than $a$
2 $\lim_{x \to a^+} f(x)$ is the right-hand limit: $x$ approaches $a$ through values greater than $a$
3 $\lim_{x \to a} f(x) = L$ exactly when both one-sided limits exist and both equal $L$
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 $a$ on the left and the piece for values just above $a$ on the right
6The value $f(a)$ (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

\frac{|x|}{x}

0

both one-sided limits exist (

-1

and

1

), 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)

. The limits come from the curve on each side as it approaches

x = a

What if the function is only defined on one side?

Then only one one-sided limit makes sense. For

\sqrt{x}

0

there are no values to the left, so we use

\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

\frac{1}{x}

grows without bound as

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 $x$ from only one side of $a$
Left-hand limit: The value $f(x)$ approaches as $x \to a^-$ , through values less than $a$
Right-hand limit: The value $f(x)$ approaches as $x \to a^+$ , through values greater than $a$
Two-sided limit: The ordinary limit $\lim_{x \to a} f(x)$ ; 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 $x$
Jump discontinuity: A break where both one-sided limits exist but are different, so the graph jumps

Formula Card

Left-hand limit

\lim_{x \to a^-} f(x)

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

Right-hand limit

\lim_{x \to a^+} f(x)

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

When the limit exists

\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

for

x > a

|x - a| = -(x - a)

for

x < a

Use this to simplify expressions with absolute values on each side

Teacher Guide: One-Sided Limits

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Parking Fees

Building a Smooth Ramp

Mobile Data Plans

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Left-hand limit

Right-hand limit

When the limit exists

Absolute value by sides

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits Graphically

Evaluating Limits from Tables