# Limit of a Function

## Limit of a Function Definition:

Behavior of a function near a particular input.

## Limit of a Function Notation:

The limit of a function ƒ(x) is L
as L approaches a

limx → a ƒ(x) = L

## Right Limit of a Function Notation:

The right limit of a function ƒ(x) is A
as x approaches a from the right

limx → a+ ƒ(x) = A

## Left Limit of a Function Notation:

The left limit of a function ƒ(x) is A
as x approaches a from the left

limx → a- ƒ(x) = A

## Limit of a Function Example:

limx → 3 2x = 6 since 2(3) = 6

## Limit of a Constant Theorem:

Given a constant c,
If ƒ(x) = c then

limx → a ƒ(x) = c

## Limit with constant multiplier Theorem:

Given a constant k:

limx → a ƒ(x)kA = k * limx → a ƒ(x)

## Limit with constant multiplier Theorem example:

limx → 3 2x = 6 since 2(3) = 6
Use the limit theorem with a multiplier:
limx → 3 2x = 2 * limx → 3 x
limx → 3 x = 3, so we have
limx → 3 2x = 2 * 3
limx → 3 2x = 6

## Limit of Sums Theorem:

limx → a [ƒ(x) + g(x)] =
limx → a ƒ(x) + limx → a g(x)

## Limit of Sums Theorem Example:

ƒ(x) = 2x and g(x) = x2
limx → 3[2x + x2] = limx → 3 2x + limx → 3 x2
[2(3) + 32] = 2(3) + 32
[6 + 9] = 6 + 9
15 = 15

## Limit of Differences Theorem:

limx → a [ƒ(x) - g(x)] =
limx → a ƒ(x) - limx → a g(x)
ƒ(x) = 2x and g(x) = x2
limx → 3[2x - x2] = limx → 3 2x - limx → 3 x2
[2(3) - 32] = 2(3) - 32
[6 - 9] = 6 - 9
-3 = -3

## Limit of Products Theorem:

limx → a [ƒ(x) * g(x)] =
limx → a ƒ(x) * limx → a g(x)
ƒ(x) = 2x and g(x) = x2
limx → 3[2x * x2] = limx → 3 2x * limx → 3 x2
[2(3) * 32] = 2(3) * 32
[6 * 9] = 6 * 9
54 = 54

## Limit of Quotients Theorem:

limx → a [ƒ(x) / g(x)] =
limx → a ƒ(x) / limx → a g(x)

## Function Calculator:

More help on functions calculator

### How does the Limit of a Function Calculator work?

This lesson walks you through what limit is, how to write limit notation, and limit theorems

### What 1 formula is used for the Limit of a Function Calculator?

1. limx → a ƒ(x) = L

For more math formulas, check out our Formula Dossier

### What 9 concepts are covered in the Limit of a Function Calculator?

constant
a value that always assumes the same value independent of how its parameters are varied
difference
the result of one of the important mathematical operations, which is obtained by subtracting two numbers
function
relation between a set of inputs and permissible outputs
ƒ(x)
input
what goes into a function
limit
the value that a function (or sequence) approaches as the input (or index) approaches some valu
limit of a function
product
The answer when two or more values are multiplied together
theorem
A statement provable using logic
variable
Alphabetic character representing a number