Calculus I · Limits and Continuity · lesson

One-Sided Limits From the Left and Right

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?

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Show hint

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

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Worked example

A piecewise function with matching sides

Let

f(x)= \begin{cases} x+1,&x<2,\\ 5-x,&x\ge2. \end{cases}

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)= \begin{cases} x+1,&x<2,\\ 6-x,&x\ge2. \end{cases}

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 $4$ .

Piecewise graph with a jump at x = 2. — **Read the graph**
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.

After the explanation

Use the section idea

Reading lens

What are nearby outputs doing as the input approaches the target from both sides?

Mental model

Imagine tightening a window around the target input and watching where all nearby outputs are forced to gather.

Decision

Read the left-hand and right-hand behavior separately first; combine them only after both sides approach the same output.

Common trap

The function value at the target can be missing or deliberately moved, so never substitute a plotted dot for evidence from both sides.

Check yourself

You understand the section when you can explain a limit from a graph, table, and sentence without confusing it with the function value.

Source & rights

Original instruction with traceable references.

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