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Concept

Limit

The limit of a function

f (x)

x

approaches

c

is the value

f (x)

to which the function keeps getting closer as

x

approaches

c .

This is typically written as follows.

x \to c lim f (x) = L

In other words, if the limit exists, the output of

f

gets arbitrarily close to

L

when its input

x

is close enough to

c

from either side. In the following example,

f (x)

keeps getting closer to

2

x

approaches

1 .

This means that the limit of

f (x)

x

approaches

1

is equal to

2 .

The limit of the function f as x approaches 1 is 2

Extra

When Do Limits Exist?

In order for a limit to exist, both one-sided limits need to exist, and they must be equal to the same value. This is formally stated as follows.

x \to c lim f (x) = L ⇕ x \to c^{-} lim f (x) = x \to c^{+} lim f (x) = L

Consider the following function.

f (x) = \frac{∣ x ∣}{x}

Its graph is shown below.

Note that as

x

approaches

0

from the left, the function keeps getting closer to

- 1 .

x \to c^{-} lim f (x) = - 1

However, when

x

approaches

0

from the right, the function keeps getting closer to

1

instead.

x \to c^{+} lim f (x) = 1

Since both one-sided limits do not coincide, the limit does not exist.

x \to c lim f (x) does not exist

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