Category: Limit of the Function

Definition of the Limit of a Function

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.

One-Sided Limits

Now we can extend concept of limit.

Definition. We write that limxaf(x)=L\lim_{{{x}\to{{a}}^{{-}}}}{f{{\left({x}\right)}}}={L} and say "the limit of f(x), as x approaches a from the left, equals L" if for any ϵ>0\epsilon>{0} there exists δ>0\delta>{0} such that f(x)L<ϵ{\left|{f{{\left({x}\right)}}}-{L}\right|}<\epsilon when xa<δ{\left|{x}-{a}\right|}<\delta and x>a{x}>{a}.

Sandwich Theorem

Fact. If f(x)g(x){f{{\left({x}\right)}}}\le{g{{\left({x}\right)}}} when x{x} is near a{a} (except possibly at a{a}) then limxaf(x)limxag(x)\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}\le\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}.

Properties of the Limits

Now it is time to give the properties of the limits that will allow to calculate the limits easier.

Suppose that cc is a constant and the limits limxaf(x)\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}} and limxag(x)\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}} exist; then, the following laws hold:

Limits Involving Infinity

Now it is time to talk about the limits that involve the special symbol \infty.

First, we are going to talk about infinite limits.

Definition. We write that limxaf(x)=\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty (limxaf(x)=)\left(\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=-\infty\right) if for any E>0{E}>{0} there exists δ>0\delta>{0} such that f(x)>E (f(x)<E){f{{\left({x}\right)}}}>{E}\ {\left({f{{\left({x}\right)}}}<-{E}\right)} when xa<δ{\left|{x}-{a}\right|}<\delta.

Asymptotes

Definition. The line x=ax=a is called vertical asymptote of the curve y=f(x)y=f{{\left({x}\right)}} if at least one of the following statements is true:

  1. limxaf(x)=\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty
  2. limxa+f(x)=\lim_{{{x}\to{{a}}^{+}}}{f{{\left({x}\right)}}}=\infty
  3. limxaf(x)=\lim_{{{x}\to{{a}}^{{-}}}}{f{{\left({x}\right)}}}=\infty
  4. limxaf(x)=\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=-\infty
  5. limxa+f(x)=\lim_{{{x}\to{{a}}^{+}}}{f{{\left({x}\right)}}}=-\infty
  6. limxaf(x)=\lim_{{{x}\to{{a}}^{{-}}}}{f{{\left({x}\right)}}}=-\infty

For example y-axis (x=0)\left({x}={0}\right) is a vertical asymptote of the curve y=1x2{y}=\frac{{1}}{{{x}}^{{2}}} because limx01x2=\lim_{{{x}\to{0}}}\frac{{1}}{{{x}}^{{2}}}=\infty.

Indeterminate Forms for Functions

In the note Limits Involving Infinity we saw that it is pretty easy to evaluate limx0x+1x\lim_{{{x}\to{0}}}\frac{{{x}+{1}}}{{x}} because since limx0(x+1)=1\lim_{{{x}\to{0}}}{\left({x}+{1}\right)}={1} and limx0x=0\lim_{{{x}\to{0}}}{x}={0} then division of 1 by very small number will give very large number, and so limx0x+1x=\lim_{{{x}\to{0}}}\frac{{{x}+{1}}}{{x}}=\infty.