1.3 - Estimating Limit Values from Graphs

Introduction to One-Sided Limits

So far we’ve looked at limits from both directions at once. Now we’ll look at one-sided limits, where we approach an

x

-value from only one direction, either the left or the right.

A one-sided limit still describes the

y

-value a function approaches as

x

approaches a given value. We just restrict which side we come from.

Notation

Left-hand limit (approaching from the left)

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

The minus sign means we’re approaching from the left side

We move toward $a$ from values less than $a$

Think: moving in the positive direction along the $x$ -axis toward $a$

Right-hand limit (approaching from the right)

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

The plus sign means we’re approaching from the right side

We move toward $a$ from values greater than $a$

Think: moving in the negative direction along the $x$ -axis toward $a$

Example: When One-Sided Limits Differ

1.3 - Estimating Limit Values from Graphs

Left-hand limit:

\lim_{x \to 3^-} f(x) = -1

As we approach

x = 3

from the left (traveling along the curve from

x < 3

), the

y

-values head toward

-1

Right-hand limit:

\lim_{x \to 3^+} f(x) = 2

As we approach

x = 3

from the right (from

x > 3

), the

y

-values head toward

2

Two-sided limit:

\lim_{x \to 3} f(x) = \text{DNE}

Since the left-hand and right-hand limits are different, the two-sided limit does not exist. For a two-sided limit to exist, both one-sided limits have to approach the same value.

The Rule for Two-Sided Limits

\lim_{x \to a} f(x) = L \text{ if and only if } \lim_{x \to a^-} f(x) = L \text{ AND } \lim_{x \to a^+} f(x) = L

In other words, the two-sided limit exists only when both one-sided limits exist and agree.

Three Ways a Limit Can Fail to Exist

A two-sided limit exists only when

f(x)

heads toward one real number from both sides. There are three common ways that breaks down:

The one-sided limits disagree. This is the example above: the left side heads to $-1$ and the right side heads to $2$ . You’ll usually see this at a jump in the graph.

The function is unbounded. As $x \to 1$ , $\frac{1}{(x - 1)^2}$ grows without bound from both sides. It isn’t approaching any real number, so the limit does not exist. (We write $\lim_{x \to 1} \frac{1}{(x - 1)^2} = \infty$ to describe how it fails. More on this in 1.14.)

The function oscillates. As $x \to 0$ , $\sin\left(\frac{\pi}{x}\right)$ keeps swinging between $-1$ and $1$ , faster and faster. It hits $1$ at $x = \frac{2}{5}, \frac{2}{9}, \frac{2}{13}, \ldots$ and $-1$ at $x = \frac{2}{3}, \frac{2}{7}, \frac{2}{11}, \ldots$ , and those points pile up right next to $0$ . It never settles on one value, so the limit does not exist.

Watch the scale. A graph only shows what fits in its window, so it can hide important behavior. A calculator graph of

y = \frac{x^2 - 4}{x - 2}

usually looks like the plain line

y = x + 2

, because the hole at

(2, 4)

is a single missing point. And no screen can draw

\sin\left(\frac{\pi}{x}\right)

accurately near

0

. If a graph looks suspicious, zoom in, or check with a table or the algebra.

Practice Problems

Find each of the following:

\lim_{x \to -2^-} f(x)

\lim_{x \to -2^+} f(x)

\lim_{x \to -2} f(x)

\lim_{x \to 1} f(x)

\lim_{x \to 0} f(x)

\lim_{x \to 3^-} f(x)

\lim_{x \to -1} f(x)

f(-2)

Solutions

\lim_{x \to -2^-} f(x) = 1

. Find

x = -2

, approach from the left (values less than

-2

), and follow the curve. It approaches

y = 1

\lim_{x \to -2^+} f(x) = -2

. Start from the right side of

x = -2

and follow the curve back toward it. The curve approaches

y = -2

\lim_{x \to -2} f(x) = \text{DNE}

. The left-hand limit is

1

and the right-hand limit is

-2

. Since

1 \neq -2

, the two-sided limit doesn’t exist.

\lim_{x \to 1} f(x) = 1

. Both sides converge to

y = 1

(at the open circle). Even though there’s a hole at

(1, 1)

and the filled dot sits at

(1, -2)

, the limit is still

1

\lim_{x \to 0} f(x) = -2

. Both sides meet at the same point,

y = -2

\lim_{x \to 3^-} f(x) = 5

. Approaching

x = 3

from the left only, the curve rises to

y = 5

\lim_{x \to -1} f(x) = -3

. Both directions approach

y = -3

f(-2) = 1

. Find the filled dot at

x = -2

. It’s at

y = 1

(ignore the open circle).

Constructing a Graph from Given Conditions

Sometimes you’ll need to sketch a function that satisfies a list of conditions involving one-sided limits, two-sided limits, and function values.

Problem: sketch a graph of

g(x)

satisfying all of the following:

$g(4) = -2$

$\lim_{x \to 4} g(x) = 3$

$\lim_{x \to -1^+} g(x) = 2$

$g$ is increasing on the interval $(-1, 4)$

$\lim_{x \to -1^-} g(x) > \lim_{x \to -1^+} g(x)$

Step 1: plot $g(4) = -2$

x = 4

, put a filled dot at

(4, -2)

Step 2: apply $\lim_{x \to 4} g(x) = 3$

x

approaches 4 from both sides, the function approaches

y = 3

. Put an open circle at

(4, 3)

. The graph has to approach this point from both left and right.

Step 3: apply $\lim_{x \to -1^+} g(x) = 2$

From the right side of

x = -1

, the function approaches

y = 2

. So we need an open circle (or some indication) at

(-1, 2)

where the graph comes in from the right.

Step 4: increasing on $(-1, 4)$

Between

x = -1

and

x = 4

, the graph has to always be rising left to right. This connects the point approaching

(-1, 2)

from the right to the point approaching

(4, 3)

from the left. Draw a smooth curve that steadily increases, starting near

y = 2

and ending near

y = 3

Step 5: apply $\lim_{x \to -1^-} g(x) > \lim_{x \to -1^+} g(x)$

The left-hand limit at

x = -1

has to be greater than the right-hand limit.

We know

\lim_{x \to -1^+} g(x) = 2

, so

\lim_{x \to -1^-} g(x) > 2

From the left side of

x = -1

, the graph must approach a

y

-value above 2. Say it approaches

y = 4

. Mark

(-1, 4)

and draw the graph coming in from the left.

Final graph

Verification

Condition 1: $g(4) = -2$ , filled dot at $(4, -2)$ ✓

Condition 2: $\lim_{x \to 4} g(x) = 3$ , both sides approach $y = 3$ ✓

Condition 3: $\lim_{x \to -1^+} g(x) = 2$ , right side approaches $y = 2$ ✓

Condition 4: increasing on $(-1, 4)$ , the graph rises throughout ✓

Condition 5: left approaches $y = 4$ , right approaches $y = 2$ , and $4 > 2$ ✓

Key Takeaways

One-sided limits look at behavior from only one direction, left or right

Notation: $\lim_{x \to a^-} f(x)$ is from the left, $\lim_{x \to a^+} f(x)$ is from the right

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

When one-sided limits differ, the two-sided limit does not exist (DNE)

A limit also does not exist when the function is unbounded or oscillates without settling on one value

Graphs can hide behavior (like a one-point hole), so zoom in or check a table when something looks off

Open circles and filled dots tell you about function values, not limits

Limits describe approaching behavior, which may differ from actual function values

When constructing graphs from conditions, work through each requirement in order and verify at the end

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