**Limit**

Previously, we found that a rational function is not defined at the zeros of the denominator.

The function *f* is not defined when *x = -2*, because then the denominator is *0*. However, we would like to know if the values of the function approach some particular number at* x = -2*. We check some of the values of the function close to the value of the variable* -2*

**x > -2**

We are closing to a value of -5

**x < -2**

We are closing to a value of -5

In both cases, we approach a value *-5*, so we can assume that when *x* approaches *-2*, the value of the function approaches *-5*.

### Finding the limit

The denominator of the function above is* 0* when *x = -2*. The numerator is also* 0* when* x = -2*. Then *f (-2)* has the form* 0/0*, which is not defined. Since they have a common zero point, we can simplify the expression. The second zero of the numerator is *3.*

**Expression **

**Expression**

The limit value of the function *f* in *a* is *b*.

**Example 1**

Find the limits

**a)**

**a)**

**b)**

**b)**

**c)**

**c)**

Solution

a) We substitute* x = 3*

The limit is *2*

a) We substitute *x = 5*

The limit is* 0*

a) We substitute *x = 4*

The value of the denominator is *0*, so the expression is not defined when* x = 4*. The value of the numerator is *-2* when* x = 4*, so we can't get a reduce the expression. The limit does not exist when *x = 4.*

At each point, the limit value was found by substituting a value of *x *into the expression. This should be done whenever you start to find a limit value.

**Example 2**

Find the limit

We substitute* x = 2*

Both the numerator and denominator are *0*, and the expression is not defined. However, this tells us that a limit value exists. The numerator and denominator have a common zero point that can be reduced out. The zero points of the numerator are *x = 2* and* x = 1,* the zero points of the denominator are* x = 2* and* x = 4*

The limit is* -1/2* when* x* approaches *2*

**Turn on the subtitles if needed**