Calculus I · Limits and Continuity · lesson

One-Sided Limits From the Left and Right

Learn One-Sided Limits From the Left and Right with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style pr

Course progress8 of 47 core pages8 of 47

Where this chapter fits

Chapter 1: What a limit means

Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.

Reading lens: What are nearby outputs doing as the input approaches the target from both sides? Keep that question in view while reading One-Sided Limits From the Left and Right; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Function Value vs. Limit to When a Limit Does Not Exist. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Read and compute left-hand and right-hand limits; use them to decide whether a two-sided limit exists.

One-Sided Limits

An input can approach $a$ from values smaller than $a$ or from values greater than $a$ .

Definition

One-Sided Limit Notation

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

means that $f(x)$ approaches $L$ as $x$ approaches $a$ using values $x<a$ .

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

means that $f(x)$ approaches $L$ as $x$ approaches $a$ using values $x>a$ .

The minus and plus signs are directions, not arithmetic operations.

Concept

Stand at the doorway $x=a$ . Approaching from the hallway on the left is $x\to a^-$ . Approaching from the hallway on the right is $x\to a^+$ . A two-sided meeting happens only if both groups arrive at the same height.

Theorem

One-Sided Test for a Two-Sided Limit

\boxed{\lim_{x\to a}f(x)=L}

if and only if

\boxed{\lim_{x\to a^-}f(x)=L \quad\text{and}\quad \lim_{x\to a^+}f(x)=L.}

If the one-sided limits disagree, the two-sided limit does not exist.

Guided walkthrough

Two staircases

Suppose the graph approaches height $3$ from the left and height $3$ from the right. Then the two-sided limit is $3$ .

If the graph approaches height $3$ from the left but height $4$ from the right, there is no single answer to "what height does the graph approach?" The two-sided limit does not exist.

Interactive checkleft-right-01

The left-hand limit is $2$ and the right-hand limit is $5$ . What is the two-sided limit?

Your work stays on this device. No account or AI grader is used.

Show hint

A two-sided limit exists only when the sides agree.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Worked example

A piecewise function with matching sides

Let

f(x)= \left\{\matrix{ x+1,\quad x<2,\quad 5-x,\quad x\ge2. }\right.

Find the left-hand, right-hand, and two-sided limits at $x=2$ .

Show worked solution

For inputs to the left of $2$ , use $x+1$ :

\lim_{x\to2^-}f(x)=2+1=3.

For inputs to the right of $2$ , use $5-x$ :

\lim_{x\to2^+}f(x)=5-2=3.

The one-sided limits agree, so

\boxed{\lim_{x\to2}f(x)=3}.

The actual value is also $f(2)=3$ , but that equality is not what made the two-sided limit exist. The matching one-sided limits did.

Worked example

A jump

Let

g(x)= \left\{\matrix{ x+1,\quad x<2,\quad 6-x,\quad x\ge2. }\right.

Find the one-sided and two-sided limits at $x=2$ .

Show worked solution

From the left,

\lim_{x\to2^-}g(x)=2+1=3.

From the right,

\lim_{x\to2^+}g(x)=6-2=4.

Because $3\ne4$ ,

\boxed{\lim_{x\to2}g(x)=\mathrm{DNE}}.

This is a jump discontinuity. The graph jumps from a left-hand height of $3$ to a right-hand height of $4$ .

Graph reading guide

Unequal one-sided limits produce a jump and no two-sided limit.

Exam note

A filled point at $x=a$ does not repair unequal one-sided limits. Even if you define $g(2)=3.5$ , the left side still approaches $3$ and the right side still approaches $4$ . No single dot can persuade two disagreeing neighborhoods to cooperate.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Calculus I · Limits and Continuity · lesson

One-Sided Limits From the Left and Right

Learn One-Sided Limits From the Left and Right with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style pr

Course progress8 of 47 core pages8 of 47

Where this chapter fits

Chapter 1: What a limit means

Build the neighborhood idea with motion, tables, holes, one-sided behavior, and graphs.

This page connects Function Value vs. Limit to When a Limit Does Not Exist. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Read and compute left-hand and right-hand limits; use them to decide whether a two-sided limit exists.

One-Sided Limits

An input can approach $a$ from values smaller than $a$ or from values greater than $a$ .

Definition

One-Sided Limit Notation

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

means that $f(x)$ approaches $L$ as $x$ approaches $a$ using values $x<a$ .

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

means that $f(x)$ approaches $L$ as $x$ approaches $a$ using values $x>a$ .

The minus and plus signs are directions, not arithmetic operations.

Concept

Theorem

One-Sided Test for a Two-Sided Limit

\boxed{\lim_{x\to a}f(x)=L}

if and only if

\boxed{\lim_{x\to a^-}f(x)=L \quad\text{and}\quad \lim_{x\to a^+}f(x)=L.}

If the one-sided limits disagree, the two-sided limit does not exist.

Guided walkthrough

Two staircases

Suppose the graph approaches height $3$ from the left and height $3$ from the right. Then the two-sided limit is $3$ .

If the graph approaches height $3$ from the left but height $4$ from the right, there is no single answer to "what height does the graph approach?" The two-sided limit does not exist.

Interactive checkleft-right-01

The left-hand limit is $2$ and the right-hand limit is $5$ . What is the two-sided limit?

Your work stays on this device. No account or AI grader is used.

Show hint

A two-sided limit exists only when the sides agree.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Worked example

A piecewise function with matching sides

Let

f(x)= \left\{\matrix{ x+1,\quad x<2,\quad 5-x,\quad x\ge2. }\right.

Find the left-hand, right-hand, and two-sided limits at $x=2$ .

Show worked solution

For inputs to the left of $2$ , use $x+1$ :

\lim_{x\to2^-}f(x)=2+1=3.

For inputs to the right of $2$ , use $5-x$ :

\lim_{x\to2^+}f(x)=5-2=3.

The one-sided limits agree, so

\boxed{\lim_{x\to2}f(x)=3}.

The actual value is also $f(2)=3$ , but that equality is not what made the two-sided limit exist. The matching one-sided limits did.

Worked example

A jump

Let

g(x)= \left\{\matrix{ x+1,\quad x<2,\quad 6-x,\quad x\ge2. }\right.

Find the one-sided and two-sided limits at $x=2$ .

Show worked solution

From the left,

\lim_{x\to2^-}g(x)=2+1=3.

From the right,

\lim_{x\to2^+}g(x)=6-2=4.

Because $3\ne4$ ,

\boxed{\lim_{x\to2}g(x)=\mathrm{DNE}}.

This is a jump discontinuity. The graph jumps from a left-hand height of $3$ to a right-hand height of $4$ .

Graph reading guide

Unequal one-sided limits produce a jump and no two-sided limit.

Exam note

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Chapter 1: What a limit means

One-Sided Limits

Related resources

Original instruction with traceable references.

Chapter 1: What a limit means

One-Sided Limits

Related resources

Original instruction with traceable references.