Transcript limits of functions

Calculus
Limits Lesson 2
Bell Activity
A. Use your calculator graph to find:
1. lim x2 (3x  2)
2. lim x3 (2 x  5)
x
3. lim x4 (4  )
2
2
( x  4)
4. lim
x  2 ( x  2)
B. Without Calculator, Find
1.
f(2) if f(x) = 3x – 2
2.
f(-3) if f(x) = 2x + 5
3.
f(4) if f(x) = 4 – x/2
4.
2
(
x
 4)
f(2) if f(x) =
( x  2)
Notice: In #4, both the numerator and
denominator are 0. This is called an
indeterminate form and thus cannot be evaluated.
The limit may be estimated or found by
creating a table of values very close to the
approaching x value.
( x  3x  2)
lim
=1
( x  2)
x 2
2
X
1.75
1.9
1.99
Y
.75
.9
.99
2
2.001
1.001
2.01
1.01
2.1
1.1
***
lim x3 (2 x  5)
=1
X
2.8
2.9
2.99
3
3.001
3.01
3.1
Y
.6
.8
.98
1
1.002
1.02
1.2
It would seem from this example that at times a
limit of a function as x approaches a particular #
can simply be obtained by finding f(that #).
What do you think allows this function to have the
limit as x approaches 3 and f(3) to be the same
whereas the first problem does not?
lim x3 (2 x  5)
x  3x  2
x2
=1
2
lim x2
=1
Which of these can be substituted directly to find
the limit?
1.
2.
3.
lim x 3 ( x 2  2) =11
x2 1
lim x 2
x 1
2
x 1
lim x 1
x 1
=3
=2
Even though # 3’s limit can’t be found by substituting
directly we can still get the limit from a graph or a
table.
Let’s go back to a problem where the
substitution would not work to find the limit.
Perhaps we could modify the problem before
we substituted.
2
lim x1
x 1
x 1
( x  1)( x  1)
lim x 1
x 1
lim x1 ( x  1)
=2
Modify these and find the limit by substituting:
x  x6
2
x 9
lim x3
x 1
x 1
3
2
lim x1
Let’s verify these with the calculator
graph.
Find the limit:
x 1
x 1
2
lim x1
Let’s verify with the calculator graph.
Find the limit:
Let’s verify with the calculator graph.
There are other times when we cannot
find the limit by either substituting or
factoring.
lim x 0
In this case, Rationalize the Numerator
x 1 1
x
And, of course, there are problems
which merely need to be simplified
before substituting.
( x  x)  x
x
3
lim x0
3
Find the limit:
Let’s verify with the calculator graph.
Find the limit:
Let’s verify with the calculator graph.
Find the limit:
Let’s verify with the calculator graph.
Properties of Limits
If L, M , c, and k are real numbers and
lim f  x   L
x c
1.
Sum Rule :
and
lim g  x   M , then
x c
lim  f  x   g  x    L  M
x c
The limit of the sum of two functions is the sum of their limits.
2.
DifferenceRule :
lim  f  x   g  x    L  M
x c
The limit of the difference of two functions is the difference
of their limits.
Slide 2- 17
Properties of Limits continued
3.
Product Rule:
lim  f  x  g  x    L M
x c
The limit of the product of two functions is the product of their limits.
4.
Constant Multiple Rule:
lim  k f  x    k L
x c
The limit of a constant times a function is the constant times the limit
of the function.
5.
Quotient Rule :
lim
x c
f  x
g  x

L
, M 0
M
The limit of the quotient of two functions is the quotient
of their limits, provided the limit of the denominator is not zero.
Slide 2- 18
Properties of Limits continued
6.
If r and s are integers, s  0, then
Power Rule :
r
s
r
s
lim  f  x    L
x c
r
s
provided that L is a real number.
The limit of a rational power of a function is that power of the
limit of the function, provided the latter is a real number.
Other properties of limits:
lim  k   k
x c
lim  x   c
x c
Example Properties of Limits
Use any of the properties of limits to find
lim  3x3  2 x  9 
x c
lim  3x3  2 x  9   lim3x3  lim 2 x  lim9
x c
x c
x c
 3c3  2c  9
x c
sum and difference rules
product and multiple rules
Assignment

Text p. 67, # 1 – 43 odds