Limits and Their Properties

1.2 Finding Limits Graphically and Numerically

Focus on what a limit describes, estimate a limit from a value table, read a limit from a graph, the limit and the function value can differ, recognize when a finite limit does not exist, connect ε and δ to the formal definition.

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Ideas in this lesson

01What a limit describesWorked steps · five questions 02Estimate a limit from a value tableWorked steps · five questions 03Read a limit from a graphWorked steps · five questions 04The limit and the function value can differWorked steps · five questions 05Recognize when a finite limit does not existWorked steps · five questions 06Connect ε and δ to the formal definitionWorked steps · five questions

Idea 1 of 6

What a limit describes

A limit describes the output approached as the input gets close to a point from both sides. The input need not equal that point.

Worked example 1

x

approaches

1

from both sides, the values of

f(x) = 2x + 1

approach

3

. Find

\displaystyle \lim\limits_{x\to 1} f(x)

$\displaystyle \lim\limits_{x\to 1} f(x) = 3$ . Both sides approach $3$ , so the limit is $3$ .

Answer: $3$

Worked example 2

x

approaches

2

from both sides, the values of

f(x) = 3x + 1

approach

7

. Find

\displaystyle \lim\limits_{x\to 2} f(x)

$\displaystyle \lim\limits_{x\to 2} f(x) = 7$ . Both sides approach $7$ , so the limit is $7$ .

Answer: $7$

Idea 2 of 6

Estimate a limit from a value table

Use inputs on both sides of the approach point. Look for a common output value; do not treat rounded data as an exact proof.

Worked example 1

Use the table to estimate

\displaystyle \lim\limits_{x\to 1} f(x)

Values near the approach point
$x$	$0.90$	$0.99$	$1.01$	$1.10$
$f(x)$	$3.80$	$3.98$	$4.02$	$4.20$

Estimated $\displaystyle \lim\limits_{x\to 1} f(x) \approx 4$ . From either side of $1$ , the listed outputs get closer to $4$ . The estimated limit is $4$ .

Answer: $4$

Worked example 2

Use the table to estimate

\displaystyle \lim\limits_{x\to 2} f(x)

Values near the approach point
$x$	$1.90$	$1.99$	$2.01$	$2.10$
$f(x)$	$4.80$	$4.98$	$5.02$	$5.20$

Estimated $\displaystyle \lim\limits_{x\to 2} f(x) \approx 5$ . From either side of $2$ , the listed outputs get closer to $5$ . The estimated limit is $5$ .

Answer: $5$

Idea 3 of 6

Read a limit from a graph

Follow the curve toward the approach point from the left and from the right. A hole does not prevent a limit when the curve approaches the same height on both sides.

Worked example 1

Read

\displaystyle \lim\limits_{x\to 1} f(x)

from the graph.

$\displaystyle \lim\limits_{x\to 1} f(x) = 3$ . The curve approaches height $3$ from both sides of $x = 1$ . The filled point at height $1$ gives $f(1)$ , not the limit.

Answer: $3$

Worked example 2

Read

\displaystyle \lim\limits_{x\to 2} f(x)

from the graph.

$\displaystyle \lim\limits_{x\to 2} f(x) = 4$ . The curve approaches height $4$ from both sides of $x = 2$ . The filled point at height $1$ gives $f(2)$ , not the limit.

Answer: $4$

Idea 4 of 6

The limit and the function value can differ

The limit depends on values near the point. The function value is the output at the point. They may agree, differ, or the function value may be undefined.

Worked example 1

For

x \ne 1

, let

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

. Define

f(1) = 7

. Find

\displaystyle \lim\limits_{x\to 1} f(x)

$f(x) = x + 1 (x \ne 1)$ ; $\displaystyle \lim\limits_{x\to 1} f(x) = 2$ . For $x \ne 1$ , factor and cancel to get $f(x) = x + 1$ . The nearby outputs approach $2$ . The assigned point value does not change the limit.

Answer: $2$

Worked example 2

For

x \ne 2

, let

f(x) = \dfrac{x^{2} - 4}{x - 2}

. Define

f(2) = 9

. Find

\displaystyle \lim\limits_{x\to 2} f(x)

$f(x) = x + 2 (x \ne 2)$ ; $\displaystyle \lim\limits_{x\to 2} f(x) = 4$ . For $x \ne 2$ , factor and cancel to get $f(x) = x + 2$ . The nearby outputs approach $4$ . The assigned point value does not change the limit.

Answer: $4$

Idea 5 of 6

Recognize when a finite limit does not exist

A finite two-sided limit fails when the two sides approach different values, outputs grow without bound, or outputs keep oscillating without approaching a single value.

Worked example 1

A graph approaches

2

from the left and

6

from the right as

x

approaches

1

. What is the two-sided limit?

$\displaystyle \lim$ ₓ $\displaystyle \to _{1}^{-} f(x) = 2 \ne 6 = \lim$ ₓ $\to _{1}^{+} f(x) \implies DNE$ . A two-sided limit requires the same value from both sides. The values $2$ and $6$ differ.

Answer: Does not exist

Worked example 2

Does

\displaystyle \lim\limits_{x\to 0} \dfrac{1}{x^{2}}

have a finite real value?

$\displaystyle \lim$ ₓ $\to _{0} \dfrac{1}{x^{2}} = \infty$ ; no finite real limit. As $|x|$ gets smaller, $\dfrac{1}{x^{2}}$ grows without bound. There is no finite real limit; later this behavior can be described using an infinite limit.

Answer: No; outputs grow without bound

Idea 6 of 6

Connect ε and δ to the formal definition

For every $\epsilon > 0$ , find a $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \epsilon$ . The output tolerance $\epsilon$ is chosen first; the input tolerance $\delta$ may depend on it.

Worked example 1

For

\displaystyle f(x) = 2x + 1, \lim\limits_{x\to 1} f(x) = 3

. Which choice of

\delta

guarantees the output error is less than

\epsilon

$|f(x) - 3| = 2|x - 1| < 2\delta = \epsilon$ . The output distance is $|f(x) - 3| = 2|x - 1|$ . Choosing $\delta = \dfrac{\epsilon }{2}$ makes it less than $\epsilon$ whenever $0 < |x - 1| < \delta$ .

Answer: $\dfrac{\epsilon }{2}$

Worked example 2

For

\displaystyle f(x) = 3x + 1, \lim\limits_{x\to 1} f(x) = 4

. Which choice of

\delta

guarantees the output error is less than

\epsilon

$|f(x) - 4| = 3|x - 1| < 3\delta = \epsilon$ . The output distance is $|f(x) - 4| = 3|x - 1|$ . Choosing $\delta = \dfrac{\epsilon }{3}$ makes it less than $\epsilon$ whenever $0 < |x - 1| < \delta$ .

Answer: $\dfrac{\epsilon }{3}$

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, § $1.2, p$ . $48$ . The lesson practice preserves the approved textbook exercise selection.

MathLab with Mr. Ahmed

1.2 Finding Limits Graphically and Numerically

Ideas in this lesson

What a limit describes

Worked example 1

Worked example 2

Estimate a limit from a value table

Worked example 1

Worked example 2

Read a limit from a graph

Worked example 1

Worked example 2

The limit and the function value can differ

Worked example 1

Worked example 2

Recognize when a finite limit does not exist

Worked example 1

Worked example 2

Connect ε and δ to the formal definition

Worked example 1

Worked example 2