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 $\lim_{{{x}\to{0}}}\frac{{{x}+{1}}}{{x}}=\infty$.