We've already talked about the limit of a sequence, and, since a sequence is a particular case of a function, there will be a similarity between the sequence and the function. We are also going to extend some concepts.
Now we can extend concept of limit.
Definition. We write that limx→a−f(x)=L and say "the limit of f(x), as x approaches a from the left, equals L" if for any ϵ>0 there exists δ>0 such that ∣f(x)−L∣<ϵ when ∣x−a∣<δ and x>a.
Fact. If f(x)≤g(x) when x is near a (except possibly at a) then limx→af(x)≤limx→ag(x).
Now it is time to give the properties of the limits that will allow to calculate the limits easier.
Suppose that c is a constant and the limits limx→af(x) and limx→ag(x) exist; then, the following laws hold:
Now it is time to talk about the limits that involve the special symbol ∞.
First, we are going to talk about infinite limits.
Definition. We write that limx→af(x)=∞ (limx→af(x)=−∞) if for any E>0 there exists δ>0 such that f(x)>E (f(x)<−E) when ∣x−a∣<δ.
Definition. The line x=a is called vertical asymptote of the curve y=f(x) if at least one of the following statements is true:
- limx→af(x)=∞
- limx→a+f(x)=∞
- limx→a−f(x)=∞
- limx→af(x)=−∞
- limx→a+f(x)=−∞
- limx→a−f(x)=−∞
For example y-axis (x=0) is a vertical asymptote of the curve y=x21 because limx→0x21=∞.
In the note Limits Involving Infinity we saw that it is pretty easy to evaluate limx→0xx+1 because since limx→0(x+1)=1 and limx→0x=0 then division of 1 by very small number will give very large number, and so limx→0xx+1=∞.