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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