Ximpledu

Limit (Math)

See how to solve the limit of a function/sequence.
31 examples and their solutions.

Convergence

Definition


limn → ∞an = α
an gets close to a constant value (α).
→ Limit of an is convergent.

limn → ∞an is read as
limit, n goes to infinity, of an.

Divergence

Divergence: an does not get to a constant value.

Diverge to ∞


limn → ∞an = ∞

Diverge to -∞


limn → ∞an = -∞

Oscillation


limn → ∞an: Oscillation
an does not go to one direction
→ Limit of an is oscillating.

Limit of a Sequence

Formula

1 = 0

Example

an = 5 + 4n2
Convergent or divergent?
Solution

Example

an = √n - 2
Convergent or divergent?
Solution

Example

an = -n2 + 1
Convergent or divergent?
Solution

Example

an = (-1)n
Convergent or divergent?
Solution

Limit of a Function

Definition

y = f(x)

limx → af(x) = α
The limit of f(x) means
where the graph of y = f(x) is going
as x → a.
It doesn't mean the function value f(a).

Example

limx → 2(x2 + 5 + 3x)
Solution

Example

f(x) = {x + 2 (x ≠ 1)
4 (x = 1)


limx → 1f(x) = ?
Solution

One-Sided Limits

Left-Hand Limit

y = f(x)

limx → a-f(x) = α
a-: The number that is little bit less than a.
(a- ≈ a)

Right-Hand Limit

y = f(x)

limx → a+f(x) = α
a+: The number that is little bit greater than a.
(a+ ≈ a)

When Does a Limit Exist?

y = f(x)

limx → a-f(x) = limx → a+f(x) = α

limx → af(x) = α
(left-hand limit) = (right-hand limit)
→ Limit exists.

Example

y = f(x)

1. limx → 3-f(x) = ?
2. limx → 3+f(x) = ?
3. limx → 3f(x) = ?
Solution

Example

f(x) = {-x + 6 (x < 2)
x2 (x ≥ 2)


limx → 2f(x) = ?
Solution

Example

f(x) = x(x + 1)|x|

limx → 0f(x) = ?
Solution

∞/∞ Form

Example

limn → ∞n2 + n8n
Solution

Example

limn → ∞9n + 2n3
Solution

Example

limn → ∞3n2 + n - 5n2 - 4n
Solution

Example

limn → ∞3n + 4 + 22n + 1 - 34n - 4⋅3n - n100
Solution

Example

limn → ∞3n - 14n2 + 5 + n2 + 1
Solution

∞ - ∞ Form

Example

limn → ∞(n2 + 6n + 12 - n)
Solution

Example

limn → ∞7n2 + n - n2 - n
Solution

0/0 Form

Example

limx → 1x2 + x - 2x - 1
Solution

Example

limx → 2x + 7 - 3x - 2
Solution

Example

limx → 3x2 + ax + bx - 3 = 1

a, b = ?
Solution

0 × ∞ Form

Example

limx → 01x (6x + 3 - 2)
Solution

Squeeze Theorem

Theorem

f(x) ≤ g(x) ≤ h(x)

If limx → af(x) = limx → 0h(x) = α,
then limx → ag(x) = α.

Example

limx → ∞sin xx
Solution

Limit of sin xx

Formula

limx → 0sin xx = 1

Example

limx → 0sin 4xx
Solution

Example

limx → 0sin (sin x)x
Solution

Example

limx → 0tan xx
Solution

Example

limx → 01 - cos xx2
Solution

Constant e

Definition

(1 + 0) = e
The constant e is a special number.
e = 2.718...
e is also called
the Euler's number, natural base, natural constant.

Formula

limx → 0(1 + x)1x = e

limx → ∞(1 + 1x)x = e

Example

limx → 0(1 + 7x)1x
Solution

Example

limx → ∞(1 + 5x)x
Solution

Limit of ln (1 + x)x

Formula

limx → 0ln (1 + x)x = 1

Example

limx → 0ln (1 + 2x)x
Solution

Limit of loga (1 + x)x

Formula

limx → 0loga (1 + x)x = 1ln a
Logarithm

Example

limx → 0log2 (1 + x)sin x
Solution

Limit of ex - 1x

Formula

limx → 0ex - 1x = 1

Example

limx → 0e6x - 1x
Solution

Limit of ax - 1x

Formula

limx → 0ax - 1x = ln a

Example

limx → 04x - 1x(2x + 1)
Solution