# Category: Limit of the Function

## Definition of Limit of the Function

We already talked about limit of the sequence and since sequence is particular case of function then there will be similarity between sequence and function. And also we will extend some concepts.

Example 1. Let's investigate behavior of y=x^2 near point x=2.

## One-Sided Limits

Now we can extend concept of limit.

Definition. We write that lim_(x->a^-)f(x)=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 |f(x)-L|<epsilon when |x-a|<delta and x>a.

## Sandwich Theorem

Fact. If f(x)<=g(x) when x is near a (except possibly at a) then lim_(x->a)f(x)<=lim_(x->a)g(x).

This fact means that if values of f(x) are not larger than values of g(x) near a, then f(x) approaches not larger limit than g(x) as x->a.

## Properties of the Limits

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

Suppose that c is a constant and the limits lim_(x->a)f(x) and lim_(x->a)g(x) exist then following laws hold:

## Limits Involving Infinity

Now it is time to talk about limits that involve special symbol oo.

First we talk about infinite limits.

Definition. We write that lim_(x->a)f(x)=oo (lim_(x->a)f(x)=-oo) if for any E>0 there exists delta>0 such that f(x)>E\ (f(x)<-E) when |x-a|<delta.

## Asymptotes

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:

1. lim_(x->a)f(x)=oo
2. lim_(x->a^+)f(x)=oo
3. lim_(x->a^-)f(x)=oo
4. lim_(x->a)f(x)=-oo
5. lim_(x->a^+)f(x)=-oo
6. lim_(x->a^-)f(x)=-oo

For example y-axis (x=0) is a vertical asymptote of the curve y=1/x^2 because lim_(x->0)1/x^2=oo.

## Indeterminate Forms for Functions

In the note Limits Involving Infinity we saw that it is pretty easy to evaluate lim_(x->0)(x+1)/x because since lim_(x->0)(x+1)=1 and lim_(x->0)x=0 then division of 1 by very small number will give very large number, and so lim_(x->0)(x+1)/x=oo.