Download lesson 6 properties of limits continued 2014

AP Calculus AB 2014-2015 Lesson 6 p. 1
Calculus Warm-Up↑ Day
1.
Given the piecewise-defined function j(x) defined below, evaluate lim j x  :
x  2
x  32  4,
j x   
 x  5 ,
A.
B.
C.
D.
-3
-7
DNE
-5

2.
Evaluate lim x 2  3x  2
3.
Evaluate lim
4.
Evaluate lim
x4

4x5  x 2
x 0
x2
x 2  5 x  36
x 9
x9
x  2 

x  2
AP Calculus AB 2014-2015 Lesson 6 p. 2
Lesson 6: Properties of Limits Continued…



Other Properties of Limits (Limit Laws) that result from lim  f ( x )g( x )  lim f ( x ) lim g( x )
x c


x c
 
7.
lim  f ( x )  lim f ( x )
8.
lim x n  c n , where n is a positive number
9.
lim n x  n c , where n is a positive number and c > 0 if n is even.
10.
lim n f ( x )  n lim f ( x )
n
xc
x c
Example: lim x 2  lim x
n
x5
x5
2
or
x c
  
lim x 2  lim x lim x
x5
x5
x5
xc
x c
xc
xc
Evaluate the limit and justify each step by indicating the appropriate Properties of Limits
(Limit Laws):
Example 1:
Example 2:
Example 3:


lim t 2  1 t  3
t 1
3
5
1  3x


lim 

2
4
x 1 1  4 x  3 x


lim u 4  3u  6
u 2
3
AP Calculus AB 2014-2015 Lesson 6 p. 3
Evaluate the following limits, if they exist:
Some helpful factors:
x2  x  6
x2
x2
Example 4:
lim
Example 5:
lim
Example 6:
x 2  2x  1
lim
x 1
x 4 1
9t
x 9 3  t
x
x
x
x
2
3
3
4

 a   a  b a  ab  b 
 a   a  b a  ab  b 
 a   ( x  a )( x  a )( x  a
 a 2  ( x  a )( x  a )
3
2
2
3
2
2
4
2
2
)
AP Calculus AB 2014-2015 Lesson 6 p. 4
Using Properties of Limits to Prove Properties of Continuous Functions
Remember that the function f is continuous at x=c if f is defined at x=c and if lim f ( x )  f ( c ) .
x c
Suppose that both f(x)and g(x) are continuous at x = c.
(1)
Show that f(x) + g(x) is also continuous at x = c.
Proof: lim  f ( x )  g( x ) 
x c
(2)
Show that f(x)g(x) is also continuous at x = c.
Proof: lim  f ( x )g( x ) 
x c
Continuity of Sums, Products, and Quotients of Functions
Suppose that f and g are continuous on an interval and that b is a constant. Then, on that
same interval,
1.
2.
3.
4.
bf(x) is continuous.
f(x)+g(x) is continuous.
f(x)g(x) is continuous.
f(x)
is continuous, provided g( x )  0.
g( x )
Continuity of Composite Functions
If f and g are continuous, and if the composite function f(g(x)) is defined on an interval
then f(g(x)) is continuous on that interval.
AP Calculus AB 2014-2015 Lesson 6 p. 5
Name _________________________________________________ Date _________________
Homework for Lesson 6: Properties of Limits Continued….
1) Given that lim f ( x )  4, lim g( x )  2, lim h( x )  0 find the limits that exist. If the limit
x 2
x 2
x 2
does not exist, explain why.
3
a)
lim g( x )
x 2
3f(x)
x2 g ( x )
b)
lim
c)
lim
d)
lim
g( x )h( x )
x 2
f(x)
x 2
f(x)
2) The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit
does not exist, explain why.
a)
lim  f ( x )  g( x )
b)
lim  f ( x )  g( x )
c)
lim  f ( x )g( x )
d)
f(x)
x 1 g ( x )
e)
lim x 3 f ( x )
f)
lim 3  f ( x )
x 2
x 1
x 0
lim
x2
x 1


AP Calculus AB 2014-2015 Lesson 6 p. 6
3. Evaluate the limit and justify each step by indicating the appropriate Properties of Limits
(Limit Laws)
a)
b)


lim 1  3 x 2  6 x 2  x 3
x8

lim 16  x 2
x4
4. Evaluate the limit, if it exits.
5.
a)
t2 9
t  3 2t 2  7t  3
b)
lim
c)
x2
x  2 x 3  8
lim
h 0
1  h 1
h
lim
Journal prompt: If p is a polynomial, show that lim p( x )  p( c )
x c