Unizor - Derivatives - One-Sided Function Limits

Notes to a video lecture on http://www.unizor.com

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: limx→a+ f(x)=L
An equivalent definition using ε-δ formulation is as follows:
∀ε>0 ∃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: limx→a− f(x)=L
An equivalent definition using ε-δ formulation is as follows:
∀ε>0 ∃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:
limx→a− f(x) = limx→a+ f(x) = L
Prove that
limx→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.