Announcements
Topics:
- finish section 4.2; work on sections 4.3 and 4.4; start
section 4.5
* Read these sections and study solved examples in your
textbook!
Work On:
- Practice problems from the textbook and assignments
from the coursepack as assigned on the course web
page (under the link “SCHEDULE + HOMEWORK”)
Strategy for Evaluating Limits
#
 
0

real #

0
0

Evaluating Limits Algebraically
Evaluate each limit or state that it does not exist.
x2
(a) lim
x 1 x  1
 1x
(b) lim
x 2 x  2
x 2
(c) lim
x 4 4  x
1 x
(d) lim
x 1 x 1
1
2


Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
x
0.1
0.01
0.001
0
-0.001
-0.01
-0.1

f(x)
undefined
Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
x
f(x)
0.1
10
0.01
0.001
0
-0.001
-0.01
-0.1

undefined
Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
x
f(x)
0.1
10
0.01
100
0.001
0
-0.001
-0.01
-0.1

undefined
Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
x
f(x)
0.1
10
0.01
100
0.001
1000
0
undefined
-0.001
-0.01
-0.1


Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
x
f(x)
0.1
10
0.01
100
0.001
1000
0
undefined
-0.001
-0.01
As x  0  , y  .

-0.1

Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
As x  0  , y  .

x
f(x)
0.1
10
0.01
100
0.001
1000
0
undefined
-0.001 -1000
-0.01
-100
-0.1
-10


Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
As x  0  , y  .
 As x  0 , y  .
x
f(x)
0.1
10
0.01
100
0.001
1000
0
undefined
-0.001 -1000
-0.01
-100
-0.1
-10


Infinite Limits
Example:
Use a table of values to
estimate the value of
1
lim
x 0 x
As x  0  , y  .
 As x  0 , y  .
1
  (D.N.E)
x 0 x
So lim
x
f(x)
0.1
10
0.01
100
0.001
1000
0
undefined
-0.001 -1000
-0.01
-100
-0.1
-10
Infinite Limits
Definition:
lim f (x)  
x a
“the limit of f(x), as x
approaches a, is infinity”
means that the values of
f(x)

(y-values) increase without
bound as x becomes closer
and closer to a (from either
side of a), but x  a.
Definition:
lim f (x)  
x a
“the limit of f(x), as x
approaches a, is negative
infinity”
means that the values of f(x)
(y-values) decrease without
bound as x becomes closer
and closer to a (from either
side of a), but x  a.
Infinite Limits
Definition:
lim f (x)  
x a
“the limit of f(x), as x
approaches a, is infinity”
means that the values of
f(x)

(y-values) increase without
bound as x becomes closer
and closer to a (from either
side of a), but x  a.
Definition:
lim f (x)  
x a
“the limit of f(x), as x
approaches a, is negative
infinity”
means that the values of f(x)
(y-values) decrease without
bound as x becomes closer
and closer to a (from either
side of a), but x  a.
Infinite Limits
Example:
Determine the infinite limit.
(a) lim x  2
x 1 x  1
(b) lim csc x
x  
Note:
Since the values of these functions
do not approach a real number L,
these limits do not exist.
Vertical Asymptotes
Definition:
The line x=a is called a vertical asymptote of the
curve y=f(x) if either
lim f (x)   or lim f (x)  
x a
 Example:
x a 
Basic functions we know that have VAs:
Limits at Infinity
The behaviour of functions “at” infinity is also
known as the end behaviour or long-term
behaviour of the function.
What happens to the y-values of a function f(x)
as the x-values increase or decrease without
bounds?
lim f (x)  ?
x 
lim f (x)  ?
x 
Limits at Infinity
Possibility:
y-values also approach infinity or - infinity
Examples:
f (x)  e
f (x)  0.01x 3
x

.
Limits at Infinity
Possibility:
y-values also approach infinity or - infinity
Examples:
f (x)  e
f (x)  0.01x 3
x

.
lim e   (limit D.N.E)
x
x 
lim  0.01x 3   (limit D.N.E)
x
Limits at Infinity
Possibility:
y-values approach a unique real number L
Examples:
3x 1
f (x) 
x 2
sin x
f (x) 
x

.
Limits at Infinity
Possibility:
y-values approach a unique real number L
Examples:
3x 1
f (x) 
x 2
sin x
f (x) 
x

.
3x 1
lim
3
x  x  2
sin x
lim
0
x  x
Limits at Infinity
Possibility:
y-values oscillate and do not approach a single value
Example:
f (x)  sin x
.
Limits at Infinity
Possibility:
y-values oscillate and do not approach a single value
Example:
f (x)  sin x
.
lim sin x D.N.E.
x 
Limits at Infinity
Definition:
Definition:
lim f (x)  L
lim f (x)  L
x 
x 
“the limit of f(x), as x
approaches , equals L”
“the limit of f(x), as x
approaches  , equals L”
means that the values of f(x)
(y-values) can be made as
 as we’d like to L by
close
taking x sufficiently large.
means that the values of f(x)
(y-values) can be made as
 as we’d like to L by
close
taking x sufficiently small.

Limits at Infinity
Definition:
Definition:
lim f (x)  L
lim f (x)  L
x 
x 
“the limit of f(x), as x
approaches , equals L”
“the limit of f(x), as x
approaches  , equals L”
means that the values of f(x)
(y-values) can be made as
 as we’d like to L by
close
taking x sufficiently large.
means that the values of f(x)
(y-values) can be made as
 as we’d like to L by
close
taking x sufficiently small.

Calculating Limits at Infinity
*The Limit Laws listed previously are still valid if
“ x a” is replaced by “ x ”
Limit Laws for Infinite Limits (abbreviated):


   
c
where c is any non-zero constant
Calculating Limits at Infinity
Theorem:
If r>0 is a rational number, then lim 1r  0.
x  x
If r>0 is a rational number such that x r is
1
defined for all x, then
lim r 
 0.
x  x



Calculating Limits at Infinity
Examples:
Find the limit or show that it does not exist.
2x 2  1
(a) lim
2
x 
4x  x
(b) lim
arctan( 3x  5)


x 
Horizontal Asymptotes
Definition:
The line y=L is called a horizontal asymptote of the
curve y=f(x) if either
lim f (x)  L or lim f (x)  L
x 
x 
Example:


Basic functions we know that have HAs:
.
Limits at Infinity
What about the limits at infinity of these
functions?
2x
(a)
e
f (x) 
10x
(b)
ln x
g(x) 
x
Which part (top or bottom) goes to infinity faster?

Limits at Infinity
y  e 2x

y  10x
y x
y  ln x



Comparing Functions That Approach  at

Suppose lim f (x)   and lim g(x)  
x 
x 
f (x)
 .
x  g(x)
1. f(x) approaches infinity faster than g(x) if lim
 


f (x)
2. f(x) approaches infinity slower than g(x) if lim
 0.
x  g(x)

3. f(x) and g(x) approach infinity at the same rate if lim f (x)  L.
x  g(x)

where L is any finite number other than 0.

Comparing Functions That Approach
 at 
The Basic Functions in Increasing Order of Speed
Function
Comments
n
with
n0
x
with
 0
ax

ae
 
Goes to infinity slowly
aln x
Approaches infinity faster for larger
n
Approaches infinity faster for larger


Note:
The constant a can be any positive number and does not
change the order of the functions.


Comparing Functions That Approach
y  ex

 at 
yx
y  x2

 
y x
y  ln x

Limits at Infinity
What about the limits at infinity of these
functions?
x 0.5
(b) g(x) 
2
10x
x
e
(a) f (x)  2
5x
Which part (top or bottom) goes to 0 faster?

Limits at Infinity
Semilog Graphs
y  ln 5  2ln x


y  0.5ln x
y  ln10  2ln x
y  x


Comparing Functions That Approach 0 at
Suppose lim f (x)  0 and lim g(x)  0.
x 
x 
f (x)
 0.
x  g(x)
1. f(x) approaches 0 faster than g(x) if lim




f (x)
2. f(x) approaches 0 slower than g(x) if lim
 .
x  g(x)

f (x)
3. f(x) and g(x) approach 0 at the same rate if lim
 L.
x  g(x)

where L is any finite number other than 0.


Comparing Functions That Approach 0 at
The Basic Functions in Increasing Order of Speed
Function
ax

Comments
n
with
n0
ae x with   0
 x 2 with   0
ae


Approaches 0 faster for larger
n
Approaches 0 faster for larger

Approaches 0 really fast

Note: Again, a can be any positive constant and this 
will not affect
the ordering.



Comparing Functions That Approach 0 at
y e
y  x 2
x


y e
x 2


Comparing Functions That Approach 0 at

y  ex
Semilog Graphs

y e
ln2ln
y  2ln
y
x x
x 2
y  x 2
y  x

Limits of Sequences
Recall:
The solution of the discrete-time dynamical
system mt 1  f (mt ) is a sequence of values of
mt for t  0, 1, 2, …


Solution:

m0, m1, m2, m3, ...
Limits of Sequences
To determine the limit of the solution to a
discrete-time dynamical system, we define an
associated function, m(t) , which is defined for
all t  R .
If this function has a limit at infinity, then the

sequence shares
this limit (although the
converse is not necessarily true).
Limits of Sequences
Example:
The solution of Mt 1  12 Mt 1 is given by
Mt  M
  2
1
0 2
t
 what will eventually happen to the
If M0 10,
concentration of methadone in the patient’s

blood?
Continuity
Intuitive idea:
A process is continuous if it
takes place without
interruptions or an abrupt
change.
Geometrically, a function is
continuous if it’s graph has
no break in it.
Continuity
Definition:
A function f is continuous
at the point x=a if f(x)
approaches f(a) as x
approaches a, i.e.
lim f (x)  f (a)
x a
Continuity
Implicitly requires 3 things:
f (x) exists
1. lim
x a
2. f (a) is defined
3. lim f (x)  f (a)
x a

If f is not continuous at a
(i.e. f fails to meet at least
one of the three conditions
above), then we say that f
is discontinuous at x=a.

Continuity
Example:
Find the discontinuities of
the function and explain
why it is discontinuous
there.
h(x) 
x 1
x 2  2x  3
Start by looking at x-values where f(x) is not defined
and then check the 3 conditions of continuity.
.

Continuity
Example:
Find the discontinuities of
the function and explain
why it is discontinuous
there.
2  x
g(x)  
 x 1
x 1
x 1
Start by looking at x-values where f(x) is changes
.from one ‘piece’ to another and then check
the 3 conditions of continuity.
Which Functions Are Continuous?
Definition:
A function is said to be continuous if it is continuous at every
point in its domain.
Basic Continuous Functions:
 polynomials
ex: f (x)  4
h(x)  x 7  2x 4 1
 rational functions


3x  4
, x 1
ex: f (x) 
1 x
 root functions
ex: f (x)  x, x  0


h(x) 
5
, xR
2
1 x
g(x)  3 x , x  R
Which Functions Are Continuous?
Basic Continuous Functions:
 algebraic functions
ex:
f (x) 
x  16
x2 1
 absolute value function

f (x)  x
 exponential and logarithmic functions
x
g(x)  log 5 x, x  0
ex: f (x)  e
 trigonometric and inverse trigonometric functions
 ex: f (x)  sin
x
g(x)  arctan x
Which Functions Are Continuous?
Combining Continuous Functions:
The sum, difference, product, quotient, and composition
of continuous functions is continuous where defined.
Example:
arctan( e x x 2 )
Determine where h(x) 
is continuous.
x 1

.
Limits of Continuous Functions
Example:
arctan(e x x 2 )
Evaluate lim
.
x 0
x 1

Note: By the definition of continuity, if a
function is continuous at x=a, then we can
evaluate the limit simply by direct substitution.
The Derivative
Recall:
The instantaneous rate of
change of the function f(x) at
x=a is
f '(a)  lim
h 0
f (a  h)  f (a)
h
(provided this limit exists).
Geometrically, this number
represents the slope of the
tangent to the curve at (a, f(a)).
mP  lim mPQ
Q P
The Derivative
Definition:
Given a function f(x), the derivative of f with
respect to x is the function f’(x) defined by
df
f (x  h)  f (x)
 f '(x)  lim
h 0
dx
h
The domain of this function is the set of all xvalues for which the limit exists.

The Derivative
Definition:
Given a function f(x), the derivative of f with
respect to x is the function f’(x) defined by
df
f (x  h)  f (x)
 f '(x)  lim
h 0
dx
h
The domain of this function is the set of all xvalues for which the limit exists.

domain( f ')  domain( f )
The Derivative
Interpretations of f’:
1. The function f’(x) tells us the instantaneous
rate of change of f(x) with respect to x for all
x-values in the domain of f’(x).
2. The function f’(x) tells us the slope of the
tangent to the graph of f(x) at every point (x,
f(x)), provided x is in the domain of f’(x).

The Derivative
Example:
Find the derivative of
f (x)  x  3
and use it to calculate the
instantaneous rate of
change of f(x) at x=1.
Sketch the curve f(x) and
the tangent to the curve
at (1,2).

The Derivative
Example:
Find the derivative of
f (x)  x 2  2x.
Sketch the graph of f(x)
and the graph of f’(x).
Relationship between f’ and f
If f is increasing on an interval (c,d):
The derivative f’ is positive on (c,d).
The rate of change of f is positive for all x in (c,d).
The slope of the tangent is positive for all x in (c,d).
If f is decreasing on an interval (c,d):
The derivative f’ is negative on (c,d).
The rate of change of f is negative for all x in (c,d).
The slope of the tangent is negative for all x in
(c,d).
Critical Numbers
Definition:
c is a critical number of f if c is in the domain of f
and either f’(c)=0 or f’(c) D.N.E.
Differentiable Functions
A function f(x) is said to be differentiable at x=a if
we are able to calculate the derivative of the
function at that point, i.e., f(x) is differentiable at
x=a if
f (a  h)  f (a)
f '(a)  lim
h 0
h
exists.

Differentiable Functions
Geometrically, a function is differentiable at a point
if its graph has a unique tangent line with a welldefined slope at that point.
3 Ways a Function Can Fail to be Differentiable:
Graphs
Example:
(a) Sketch the graph of f (x)  x 2  6x .
(b) By looking at the graph of f,
sketch the graph of f’(x).

(c) Find a formula for f’(x).
Relationship Between
Differentiability and Continuity
If f is differentiable at a, then f is continuous at a.