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.


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

Example 1

Find the limits





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

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