Lesson 15-1 Limits

Objective: To calculate limits of polynomials and rational functions algebraically

To evaluate limits of functions using a calculator

Definition of Limit

If f(x) becomes arbitrarily close to a unique number L as x approaches a from either side, the limit of f(x) as x approaches a is L.

This is written as:

.)(lim Lxfax

Limits That Fail to Exist

There are three conditions under which limits do not exist:

1. The function approaches a different number coming from the right hand side as opposed to the left hand side.

2. The function heads off to pos./neg. infinity.3. The function oscillates between two fixed

values as x approaches a.

Indeterminent FormIf a function is continuous at a, then

In this case you take the a in the limit and substitute it into the function. If you get a number that is the limit. If you get 0/0 or #/0 you have to use some other method to find the limit.

)()(lim afxfax

The limit of this function as x approaches 1 is 1.

Evaluate the limit)101062(lim 23

2

xxx

x

This function is also continuous so plugging in 2 will give you the limit of the function.

Example

• Find the limit: x

xx

3sinlim

03sinlim x

xx

• When the function is not continuous at the x-value in question, it is more difficult to evaluate. If you factor either the top or the bottom or both of the rational polynomial and then cancel. You can then use direct substitution to solve the limit.

• This function does not have a value at x = 3, but you can see from the graph that as you approach 3 from both sides the value approaches 2.)(lim

3xf

x

Evaluate the limit

4

2lim

2

2

2

x

xxx )2)(2(

)2(lim2

xx

xxx )2(

lim2

x

xx

2

1

Evaluate the limit

1

1lim

2

4

1

x

xx

Multiplying by the conjugate

4

16lim16

x

xx

Evaluating on the calculator

• When the function is not continuous and it is not factorable it can be evaluated using the graphing calculator. – Enter the function in Y=– then [2nd][TBLSET]– change the independent variable to ASK– then in the [TABLE] you can enter values that

approach the x from either side and see what the limit is.

Evaluate the limit

x

xx

1coslim

0

This function is undefined at 0.

Enter values into table: .1 -.1.01 -.01.001 -.001

The limit approaches 0.

Evaluate the limit using the calculator

x

xx sin

)1ln(lim

0

Special Cases

1sin

lim0

x

xx

0cos1

lim0

x

xx