Download Lesson 15-1 Limits

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 f ( x)  L.
xa
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 Form
If a function is continuous at a, then
lim f ( x)  f (a )
xa
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.
The limit of this
function as x
approaches 1 is 1.
Evaluate the limit
lim(2 x  6 x  10 x  10)
3
2
x2
This function is also continuous so plugging
in 2 will give you the limit of the function.
Example
• Find the limit:
sin 3 x 0
lim

x 
x

sin 3 x
lim
x 
x
• When the function is not continuous at the xvalue 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.
lim f ( x)
x3
• 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.
Evaluate the limit
x2  2x
lim 2
x2 x  4
x ( x  2)
 lim
x  2 ( x  2)( x  2)
1

2
x
 lim
x  2 ( x  2)
Evaluate the limit
x4 1
lim 2
x 1 x  1
Multiplying by the conjugate
x  16
lim
x 16
x 4
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
cos x  1
lim
x 0
x
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
ln( x  1)
lim
x 0
sin x
Special Cases
sin x
lim
1
x 0
x
1  cos x
lim
0
x 0
x