Limits from Graphs and Tables

The definition looks at inputs near

a

and never at

a

itself. The value

f(a)

can equal

L

, differ from

L

, or not exist, and none of these changes the limit.

Interactive: One-Sided Limits

The function below follows one rule to the left of

x = 2

and a different rule to the right. Each rule gives its own one-sided limit at

2

, and the limit exists only if the two are equal.

When a limit does not exist

A limit fails to exist in three common ways, one in each picture below.

A jump at x = 1: the one-sided limits are 2 and 0.5.

y = 1/x² grows without bound near 0.

y = cos(π/x) oscillates between −1 and 1 near 0.

In the first picture both one-sided limits exist, but they're different numbers. In the second,

\frac{1}{x^2}

grows without bound as

x

approaches

0

from either side. We write

\lim_{x \to 0} \frac{1}{x^2} = \infty

to say so, and the limit still doesn't exist, since

\infty

isn't a number.

In the third,

\cos\frac{\pi}{x}

takes every value from

-1

1

on every interval around

0

, however short. Its values never settle on one number.

Reading limits from a graph

Example 1

The graph of

f

is shown below. Use it to find each limit or value, or explain why it doesn't exist.

The graph of f on [−5, 7]. The dashed line x = 4 is a vertical asymptote.

\begin{aligned} &\text{(a)}\;\; \lim_{x \to -2} f(x) \text{ and } f(-2) \\[8pt] &\text{(b)}\;\; \lim_{x \to 0} f(x) \text{ and } f(0) \\[8pt] &\text{(c)}\;\; \lim_{x \to 1^-} f(x),\; \lim_{x \to 1^+} f(x),\; \lim_{x \to 1} f(x) \\[8pt] &\text{(d)}\;\; \lim_{x \to 4} f(x) \end{aligned}

(a) From both sides the graph approaches the open circle at height

1

, so

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

. The filled dot above the circle gives

f(-2) = 3

. The two numbers differ, which the definition allows.

(b) The graph has a corner at

x = 0

but no break. Both sides approach

3

, so

\lim_{x \to 0} f(x) = 3

, and

f(0) = 3

as well.

(c) From the left the graph approaches the filled dot at height

2

, and from the right it approaches the open circle at height

-1

. So the one-sided limits are

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

and

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

. They differ, so

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

doesn't exist.

(d) On both sides of the asymptote the graph rises without bound, so

\lim_{x \to 4} f(x) = \infty

. The limit doesn't exist as a number, and

\infty

says how it fails.

Interactive: Limit vs. Function Value

Example 1(a) compared a limit with a function value. In the graph below, the black dot marks

f(1)

, and the pieces on either side of it determine the limit at

1

Example 2

Sketch the graph of a function

f

with all of these properties.

\begin{aligned} &\lim_{x \to -1} f(x) = 2 \\[8pt] &f(-1) \text{ is not defined} \\[8pt] &\lim_{x \to 3^-} f(x) = 1, \quad \lim_{x \to 3^+} f(x) = 4 \\[8pt] &f(3) = 4 \end{aligned}

Many graphs work. In the one below, the curve near

-1

has a single gap, an open circle at

(-1, 2)

. Its height approaches

2

from both sides, with no value at

-1

. At

x = 3

it arrives at an open circle at height

1

from the left, and leaves from a filled dot at height

4

to the right.

One function with the listed properties.

A graph is drawn at one scale, and behavior smaller than that scale can be invisible. A calculator graph of

y = \frac{x^2 - 2x - 3}{x - 3}

looks like the whole line

y = x + 1

, because the hole at

(3, 4)

is narrower than a pixel. A graph can also flatten a sharp spike or a fast oscillation. When a graph suggests a limit, check it against a formula or a table if you have one.

Reading limits from a table

A table lists values of a function at a few inputs on each side of

a

. It suggests a limit but can't prove one, since it says nothing about the inputs between its entries.

Example 3

The table gives values of three functions

F

G

, and

H

at inputs near

-3

, with

F

rounded to three decimal places. None of them is defined at

-3

. For each function, estimate the two one-sided limits at

-3

and the limit at

-3

, if they exist.

$x$	$F(x)$	$G(x)$	$H(x)$
$-3.1$	$12.410$	$-4.1$	$10^2$
$-3.01$	$12.040$	$-4.01$	$10^4$
$-3.001$	$12.004$	$-4.001$	$10^6$
$-2.999$	$11.996$	$3.999$	$10^6$
$-2.99$	$11.960$	$3.99$	$10^4$
$-2.9$	$11.610$	$3.9$	$10^2$

(a) Below

-3

the values of

F

come down toward

12

, and above

-3

they come up toward

12

. Both one-sided limits appear to be

12

, so the table suggests

\lim_{x \to -3} F(x) = 12

(b) The values of

G

approach

-4

from the left and

4

from the right. The one-sided limits appear to be

\lim_{x \to -3^-} G(x) = -4

and

\lim_{x \to -3^+} G(x) = 4

. They differ, so the table suggests that

\lim_{x \to -3} G(x)

doesn't exist.

(c) Each step closer to

-3

multiplies

H(x)

100

, on both sides, and the values are all positive. They increase without bound, so both one-sided limits appear to be

\infty

, and the table suggests that

\lim_{x \to -3} H(x) = \infty

. As with

\frac{1}{x^2}

0

, the limit doesn't exist as a number.

Example 4

The table gives values of

f(x) = \cos\frac{\pi}{x}

at six inputs near

0

. They suggest that

\lim_{x \to 0} f(x) = -\frac{1}{2}

. Show that this limit doesn't exist.

$x$	$f(x)$
$-0.3$	$-\frac{1}{2}$
$-0.03$	$-\frac{1}{2}$
$-0.003$	$-\frac{1}{2}$
$0.003$	$-\frac{1}{2}$
$0.03$	$-\frac{1}{2}$
$0.3$	$-\frac{1}{2}$

x = 0.3

0.03

, and

0.003

the input to cosine is

\frac{10\pi}{3}

\frac{100\pi}{3}

, and

\frac{1000\pi}{3}

. Each is an odd multiple of

\pi

plus

\frac{\pi}{3}

, so cosine gives

-\cos\frac{\pi}{3} = -\frac{1}{2}

. Cosine is even, so the negative inputs give the same values. Now take

x = \frac{1}{4}

\frac{1}{34}

\frac{1}{334}

, and so on. The input to cosine is

4\pi

34\pi

334\pi

, and

\begin{aligned} \cos 4\pi &= \cos 34\pi \\ &= \cos 334\pi = 1. \end{aligned}

Each of those inputs sits just beside one in the table, and they approach

0

too. So

f(x)

equals

-\frac{1}{2}

and also equals

1

at inputs as close to

0

as we like. It can't approach a single number, and the limit doesn't exist. It's the third picture under "When a limit does not exist".

One limit in three forms

Example 5

The function

p

is defined for

x \ne 0

and

x \ne 5

p(x) = \frac{x^2 + x - 30}{x^2 - 5x}.

The table gives values of

p

near

5

, rounded to four decimal places, and its graph is shown. Find

\lim_{x \to 5} p(x)

from the table, from the graph, and from the formula, and say what each one shows at

x = 5

$x$	$p(x)$
$4.9$	$2.2245$
$4.99$	$2.2024$
$4.999$	$2.2002$
$5.001$	$2.1998$
$5.01$	$2.1976$
$5.1$	$2.1765$

The graph of p for x > 0, with an open circle at (5, 2.2).

The table. The values of

p

approach

2.2

from both sides, so the table suggests the limit is

2.2

. It has no row for

x = 5

The graph. The curve approaches the open circle at

(5, 2.2)

from both sides, so the limit is

2.2

. The circle is open, and there's no filled dot above or below it, so

p(5)

isn't defined.

The formula. Substituting

5

gives

\frac{0}{0}

, which says nothing yet. Factor instead. For

x \ne 5

p(x) = \frac{(x + 6)(x - 5)}{x(x - 5)} = \frac{x + 6}{x},

and as

x

approaches

5

\frac{x + 6}{x}

approaches

\frac{11}{5} = 2.2

. So

\lim_{x \to 5} p(x) = 2.2

, as the table and the graph showed, and the formula itself is undefined at

5