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

Sandwich Theorem

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

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 $$$c$$$ is a constant and the limits $$$\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}$$$ and $$$\lim_{{{x}\to{a}}}{g{{\left({x}\right)}}}$$$ 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 $$$\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=\infty$$$ $$$\left(\lim_{{{x}\to{a}}}{f{{\left({x}\right)}}}=-\infty\right)$$$ if for any $$${E}>{0}$$$ there exists $$$\delta>{0}$$$ such that $$${f{{\left({x}\right)}}}>{E}\ {\left({f{{\left({x}\right)}}}<-{E}\right)}$$$ when $$${\left|{x}-{a}\right|}<\delta$$$.

Asymptotes

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

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

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

Indeterminate Forms for Functions

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