Limits - maths sem 1 first partial

What is the mathematical way of defining a ‘donut’ neigbourhood for f : A ⊆ ℝ → ℝ

Bε(x₀) ∖ {x₀} = (x₀ − ε, x₀) ∪ (x₀, x₀ + ε)

basically the neighbourhood radius sigma around the point x₀, including every point within the neighbourhood except x₀ itself.

What is x₀ in that definition?

A limit point / accumulation point of A

What is ε (epsilon)?

How close we want f(x) to be to L

Looks at the vertical distance on the y axis

What is δ (delta)?

How close x must be to x₀

Looks at the horizontal distance on the x-axis

What is the ε-δ definition of a finite limit of a function?

lim(x → x₀) f(x) = L

if and only if:

∀ ε > 0, ∃ δₑ > 0 such that ∀ x ∈ A:

0 < |x − x₀| < δₑ ⇒ |f(x) − L| < ε

Where x₀ is a limit point of the domain A.

What does the ε-δ definition of a limit mean intuitively?

For any desired closeness ε of f(x) to L, however small, we can find a distance δ such that whenever x is within δ of x₀ (but not equal to x₀), f(x) is within ε of L.

ε = Maximum allowed distance between f(x) and L (vertical distance).

δ = Maximum allowed distance between x and x₀ (horizontal distance).

so if u pick any epsilon, u can then look at the graph and pick any delta that means that if the graph is within delta of the limit point, f(x) is within epsilon of the limit point

Why does the definition of a limit require 0 < |x − x₀| rather than simply |x − x₀| < δ?

Because a limit concerns the behaviour of f(x) as x approaches x₀, not necessarily its value at x₀.

Therefore, the limit may exist even if f(x₀) is undefined or f(x₀) ≠ L.

What is the general definition of the limit of a function using neighbourhoods?

Let f: A ⊆ ℝ → ℝ

Let x₀ be a limit point for A in ℝ̄.

lim(x → x₀) f(x) = L ∈ ℝ̄ if:

∀ V(L), ∃ U(x₀) such that ∀ x ∈ A:

x ∈ U(x₀) \ {x₀} ⇒ f(x) ∈ U(L)

In words: For every neighbourhood of L, there exists a neighbourhood of x₀ such that every x in A belonging to that neighbourhood (excluding x₀) has f(x) belonging to the neighbourhood of L.

∀ V(+ infinity) means

∃ U(+ infinity) means

Practice writing theorems: Write the theorem for when L = + infinity, x₀ is an element of R

Practice writing theorems: Write the theorem for when L is an element of R, x₀ is an element of + infinity