# Limit of a function

Firstly, we will observe the limit of a function on a intuitive level. Consider the function

The domain of the function above is , however, the value of the function can be calculated at any number which is ˝close˝ to .

From two tables above we can see that when the variable approaches , the value of the function approaches .

We can simplify the given function and draw it in the coordinate plane:

When the variable tends to , then we say that is the limit of the given function .

Let be the sequence of the numbers which are located in the domain of the given function . With this sequence is connected the sequence of the numbers . That is, if we assume that the sequence approaches the number , then we observe what is happening with the sequence .

The following definition connects the limit of a sequence and the limit of a function.

Let be an open interval and . We say that a function has a limit at the point if for every sequence in the following is valid:

We write:

We can see that the convergence of the number to is equivalent to convergence of all sequences to .

One sided limits

If we consider only sequences for which , , then we say that the variable approaches from the left and we write . This means that the difference is negative. We write

On the other side, if we consider sequences for which , then we say that variable approaches from the right and we write . This means that the difference is positive. We write

Therefore, we can conclude that the limit of the function at the point exists if:

1.) exists the limit from the left,

2.) exists  the limit from the right,

3.) limits from the left and from the right are equal.

Cauchy definition

The Cauchy definition of the limit of a function is independent of sequences.

Let be an open interval, and . The limit of a function at the point exists and if and only if

Cauchy definition of one sided limits

Let be an open interval and . The limit of the function at the point exists and if and only if

Let be an open interval and . The limit of the function at the point exists and if and only if

The limit of the constant function is equal to:

The limit of the identity function is equal to:

The limit of a power is equal to:

The limit of the th root is equal to:

Properties of the limit of a function

Let be an open intervall, and and two functions which have the limit at the point , that is:

Then the following properties are valid:

1.)

2.) the function has the limit at the point and

3.)

4.)

Example 1. Evaluate

Solution:

We know , , and by using the property 1.) we have

Example 2. Evaluate

Solution:

Since we can use the property 3.):

Example 3. Evaluate

Solution:

Now we will show that

The function is an odd function. Namely,

that is Therefore, we will consider values of the function only for positive values of the variable .

On the unit circle we draw an angle , where , .

The area of a sector () enclosed by two radii and arc  is bigger than the area of a triangle , and smaller than the area of a triangle , that is

The triangle is an isosceles triangle with arms which lengths are equal to and   the angle between them . The length of the arc is then equal to . A triangle is a right triangle where and . Therefore

The first inequality gives us

From the second inequality we have:

Finally:

When the variable approaches , approaches . Therefore, also approaches when the variable approaches :

Example 4. Evaluate

Solution:

Since and from we have

By the substitution we have:

As we showed,  , therefore

Limits at infinity

A function has the limit at point if for every sequence in the following is valid:

We write:

Similarly,

Example 5. Evaluate

Solution:

The highest power of appearing in the denominator is . Therefore, we need to divide both the numerator and denominator by . It follows

Infinite limits

Let be an open interval, and . The function has the limit at point if for every sequence in the following is valid:

Similarly,

The properties of limits specified for limits of functions at point, also apply to limits at infinity and infinite limits.

Example 6.  Consider the function , .

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