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MCV 4UI - Unit 1: Day 4
Date:______________ Section 3.3 – The Limit of a Function
Warm-up
1. Graph f ( x ) 2 x 4. 2. Use the graph to evaluate the following limit.
lim ( )
3
f x
x o
3. Evaluate the same limit algebraically.
Ex. 1. Evaluate the following limits by using the direct substitution technique.
a) lim ( 2 4 )
2
1
o
x x
x
b)
lim
2
o x
x
c) lim 9
2
3
o
x
x
Evaluating the Limit of a Function Using One-Sided Limits
1. Graph
x if x
if x
x if x
f x
2. Use the graph to evaluate one-sided limits ,
in order to find the indicated limit if it exists.
lim ( )
1
f x
x o
Left-sided limit Right-sided limit
lim ( )
1
f x
x
o
lim ( )
1
f x
x
o
3. Evaluate the same limit algebraically ,
using one-sided limits.
lim ( )
1
f x
x
o
lim ( )
1
f x
x
o
x
y
x
y
If lim f ( x ) lim f ( x )
x a x a
o o
z , then lim f ( x )
x o a
does not exist.
If lim f ( x ) L lim f ( x )
x a x a
o o
, then f x L
x o a
lim ( ).
Ex. 2. By evaluating one-sided limits , find the indicated limit if it exists. Graph the function
and state whether the function is continuous.
Note: A function is continuous if there is no break in the graph.
a)
t
2
x if x
x if x
f x ; lim ( )
2
f x
x o
b)
t
3
x if x
x if x
g x ; lim ( )
1
g x
x o
x
y
x
y
Ex. 3. Sketch the graph of any function that satisfies the given conditions in each case.
a) lim ( ) 3 , ( 3 ) 0
3
o
f x f
x
b) lim ( ) 1 , lim ( ) 1
2 2
o o
f x f x
x x
y y
x x
Ex. 4. The function p ( t )describes the production
of unleaded gasoline in a refinery, in
thousands of litres, where the time t ,
is measured in days.
a) Evaluate lim ()
1
pt
t o
b) Evaluate lim ()
3
p t
t
o
and lim ()
3
p t
t
o
c) When was the refinery shut down for repairs and when did production begin again?
d) At what times is the production function p ( t )discontinuous?
e) At what time was the production highest and what was the rate of change of production
at this time?
HW: p. 98 #4 to 8, 10 to 14
MCV 4UI - Unit 1: Day 5
Date:______________ Warm-up
1. Evaluate each limit, if possible, using the direct substitution technique for evaluating limits.
a)
2
4
lim 2
o
x
x
b) ¸
o
x
x
lim 1 1
0
c)
2 2
2
lim
x a
x a
x a
o
d)
lim
2
o x
x
e) lim ( )
4
f x
x o
if
d
if x
x
x if x
f x e) lim ( )
2
1
f x
x o
if
d
if x
x
x if x
f x
2. By evaluating one-sided limits , find the indicated limit if it exists.
Graph the function and state whether the function is continuous.
d
if x
x
x if x
f x ; lim ( )
2
f x
x o
3. Sketch the graph of any function that satisfies the given conditions in each case.
a) lim ( ) 2 , ( )
3
f x f x
x
o
is discontinuous at x 3 b) lim ( ) 1 , ( )
0
g x g x
x o
is continuous at x 0
y y
x x
x
y
Ex. 2. Evaluate the following limit using the change of variable technique.
lim
6
1
1
o x
x
x
Ex. 3. Evaluate the following limit using the one-sided limits technique.
Illustrate your results graphically.
lim
2
o x
x x
x
HW: p. 105 #1, 4, 7 to 10
x
y
MCV 4UI - Unit 1: Day 6
Date:______________ Section 3.4 – Properties of Limits
The Limit Laws
Suppose that the limits lim f ( x )
x o a
and lim g ( x )
x o a
both exist and c is a constant.
Then, we have the following limit laws.
Difference of squares Difference of cubes Sum of cubes
2 2
a b
3 3
a b
3 3
a b
Ex. 1. If lim ( ) 9
o 2
f x
x
, state and use the properties of limits to evaluate each limit.
a)
2
2
lim 2 [ f ( x )]
x o
b)
2 2 ( )
lim
f x x
f x
x
o
1. c c
x o a
lim
2. x x
x o a
lim
3. lim [ f ( x ) g ( x )] lim f ( x ) lim g ( x )
x o a x o a x o a
-The limit of a sum is the sum of the limits.
4. lim [ f ( x ) g ( x )] lim f ( x ) lim g ( x )
x o a x o a x o a
-The limit of a difference is the difference of the limits.
5. lim [ cf ( x )] c lim f ( x )
x o a x o a
- The limit of a constant times a function is the constant
times the limit of the function.
6. lim [ f ( x ) g ( x )] lim f ( x )lim g ( x )
x o a x o a x o a
- The limit of a product is the product of the limits. 7. lim ( ) 0
lim ( )
lim ( )
lim z
o
o
o
o
if g x
g x
f x
g x
f x
x a
x a
x a
x a
- The limit of a quotient is the quotient of the limits.
n
x a
n
x a
f x f x
o o
lim [ ( )] lim ( ) -The limit of a power is the power of the limit.
n n
x a
x a
o
lim
10. n
x a
n
x a
lim f ( x ) lim f ( x )
o o
- The limit of a root is the root of the limit.
n n
x a
x a
o
lim
MCV 4UI - Unit 1: Day 7
Date:______________ Section 3.5 – Continuity
When we talk about a function being continuous at a point , we mean that the graph passes through
the point without a break.
A graph that is not continuous at a point (sometimes referred to as being discontinuous at a point )
has a break of some type at the point. The following graphs illustrate these ideas.
a. Continuous for all values of the domain b. Discontinuous at x 1
(removable discontinuity)
c. Discontinuous at x 1 d. Discontinuous at x 1
(jump discontinuity) (infinite discontinuity)
Ex. 1. Given
t
2
x if x
x if x
f x ,
a) graph the function.
b) determine lim ( )
1
f x
x o
c) determine f ( 1 ).
d) is f continuous at x 1? Explain.
x x
x
y
A function f is continuous at x a if all three of the following conditions are satisfied:
- f ( a )is defined • lim f ( x )
x o a
exists • lim f ( x ) f ( a )
x o a
Ex. 2. Test the continuity of each of the following functions at x 2.
If the function is not continuous at x 2 , give a reason why it is not continuous.
Illustrate graphically.
a)
t
2
if x
x if x
g x
b)
2
x
f x
c) , 2
2
z
x
x
x x
h x and h ( 2 ) 2
HW: p. 1 10 #1, 4 to 8, 10 to 13
Review for Test: p. 115 #1 to 12, 13 to 15 (Evaluate using factoring or rationalizing techniques), 16 to 18;
Day 6 Worksheet #1 to 3 even parts; p. 119 #4, 9, 11 to 17
x
y
x
y
x
y
