Unizor - Function Limits - One-Sided Limits

Unizor - Creative Minds through Art of Mathematics - Math4Teens

One-sided Function Limit

Definition 1
Real number L is the limit of function f(x) from the right (or is the right limit) as argument x approaches real number a if for any sequence {xn}, that approaches a while each element of this sequence is greater than a, the sequence {f(xn} converges to L.
Symbolically, it looks like this: lim{x→a+} f(x)=L
An equivalent definition using ε-δ formulation is as follows:
∀ε (positive) ∃δ:
x∈(a,a+δ) ⇒ |f(x)−L| ≤ ε

Similar definition exists for the limit from the left.
Definition 2
Real number L is the limit of function f(x) from the left (or is the left limit) as argument x approaches real number a if for any sequence {xn}, that approaches a while each element of this sequence is less than a, the sequence {f(xn} converges to L.
Symbolically, it looks like this: lim{x→a−} f(x)=L
An equivalent definition using ε-δ formulation is as follows:
∀ε (positive) ∃δ:
x∈(a−δ,a) ⇒ |f(x)−L| ≤ ε

Theorem
If function f(x) converges to L as x→a, then this function converges to the same L as x→a+ or x→a−.

Proof
Both one-sided limits are supposed to be the same as a general limit. This follows from the fact that if f(xn)→L for any sequence of arguments {xn} approaching a, the same limit would be if arguments approach a only from the right or only from the left.

The converse statement is not, generally speaking, true.
For example, consider a function that is equal to 0 for all negative arguments and is equal to 1 for positive or zero arguments. This function has limit from the left 0 and limit from the right is 1.

However, if both one-sided limits exist and equal to each other, the general limit also exists and equal to these one-sided limits.

Theorem
Assume the following:
lim{x→a−} f(x) = lim{x→a+} f(x) = L
Prove that
lim{x→a} f(x) = L

Proof
Choose any positive constant ε.
Then we know that
∃δ1:x∈(a−δ1,a) ⇒ |f(x)−L| ≤ ε
and
∃δ2:x∈(a,a+δ2) ⇒ |f(x)−L| ≤ ε
Let δ=MIN(δ1,δ2).
Then both above conditions are met for this δ and we can state that
∃δ:x∈(a−δ,a+δ) ⇒ |f(x)−L| ≤ ε
which is the definition of a general limit at point x=a.

[ZorShekhtman]

It has been proven that studying Mathematics makes people more creative, more logical, simply smarter. Please share this video with your friends to make humanity smarter.

This lecture is a part of the educational Web site UNIZOR.COM dedicated to presentation of advanced course of Mathematics and Physics for teenagers interested in studying these subjects as tools to develop their creativity, analytic thinking, logic and intelligence.