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) = 6Use 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) = x2limx → 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)] =ƒ(x) = 2x and g(x) = x2
limx → a ƒ(x) - limx → a g(x)
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)] =ƒ(x) = 2x and g(x) = x2
limx → a ƒ(x) * limx → a g(x)
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)
