(Section 2.4: Limits and Infinity II) 2.4.6
Finding VAs for Graphs of Rational Functions (Expressed in Simplified Form)
If:
()
()
( ),
D ( x)
Nx
f x is rational and written in the form f x =
()
()
D ( x ) 0 (i.e., the zero polynomial), and
N ( x ) and
(Section 2.3: Limits and Infinity I) 2.3.32.
9. When dominance fails us, II. Consider lim
(
sin x +
)=
x
(
sin x +
sin x
) . We obtain:
sin x
(by Sum ID or Unit Circle) = 1 . If we had replaced x +
x
x
sin x
sin x
with x in the argument of sin x + , we wo
(Chapter 1: Review) 1.16
PART H: DOMAIN AND RANGE FROM GRAPHS
The domain of f , which is the set of all legal inputs, is the set of all
x-coordinates picked up by the graph of f . (We assume x is the independent
variable.)
(Think of crushing, or projectin
(Section 2.3: Limits and Infinity I) 2.3.31
The limit of this sequence (as the number of digits of
8.82498 . However, defining
( 2)
. We are looking for a 10th root of
( 2)
It turns out that 2
( 2)
( 2)
( 2)
31
10
approaches
3.1
=
31
has ten distinct 10th
(Chapter 1: Review) 1.15
PART G: THE VERTICAL LINE TEST (VLT)
An equation in x and y describes y as a function of x, and we can then say
y= f x
()
Its graph passes the VLT in the standard xy-plane, meaning that there is no
vertical line that intersects th
(Section 2.3: Limits and Infinity I) 2.3.30
The following variation of the contrapositive of the preceding if-then statement is also true:
1
If D x does not approach a real nonzero constant, then
does not approach a real
Dx
()
()
nonzero constant, either.
(Chapter 1: Review) 1.14
Here is the graph of f (with a few points indicated):
Warning: Clearly indicate any endpoints on a graph, such as the
origin here.
The lack of a right endpoint on our graph implies that the graph
extends beyond the edge of our fig
(Section 2.3: Limits and Infinity I) 2.3.29
FOOTNOTES
1. Infinity.
Infinity is not a number in the usual real number system that we will study in calculus.
The affinely extended real number system, denoted by
,
or
, includes two points
of infinity, one
(Chapter 1: Review) 1.13
PART F: GRAPHS OF FUNCTIONS
()
The graph of f , or the graph of y = f x , in the standard xy-plane consists of all
( ( )
points [representing ordered pairs] of the form x , f x , where x is in the domain
of f .
In a sense: Graph o
(Section 2.3: Limits and Infinity I) 2.3.28
Then,
()
V (t )
()
St
Ct=
0.6t
10 + 2t
6t
=
100 + 20t
3t
=
50 + 10t
=
Multiply by 10, though 5 is better.
Multiply by 10, though 5 is better.
in
lbs
gal
()
b) Find lim C t , and interpret the result. Discuss the
(Chapter 1: Review) 1.17
PART I: FUNCTIONS THAT ARE EVEN / ODD / NEITHER; SYMMETRY
A function f is even
f
( x) = f ( x)
x
()
Dom f
for every x in the
domain of f
()
The graph of y = f x is symmetric
about the y - axis.
Example
()
If f x = x 2 , then f is
(Section 2.4: Limits and Infinity II) 2.4.1
SECTION 2.4: LIMITS AND INFINITY II
PART A: EXPLODING GRAPHS:
VERTICAL ASYMPTOTES (VAs) and INFINITE LIMITS AT A POINT
()
In Section 2.1, we discussed lim f x , which is referred to as a limit at a point,
x
a
th
(Chapter 1: Review) 1.22
PART K: COMPOSITIONS OF FUNCTIONS
These arise when we apply a sequence of functions.
Let f and g be functions. The composite function f g is defined by
( f g ) ( x ) = f ( g ( x ) .
cfw_
Its domain is x
Rx
()
( ) .
()
Dom g and g
(Section 2.4: Limits and Infinity II) 2.4.5
Solution
x
()
lim + f x = lim +
4
x
4
x +1
x + 4x
2
3
= lim +
x
4
x +1
( x + 4)
x
4
Limit Form
3
0
0+
Regarding the denominator: Remember that
negative times positive equals negative.
=
x
()
lim f x = lim
4
x
4
(Chapter 1: Review) 1.21
PART J: ARITHMETIC COMBINATIONS OF FUNCTIONS
Let f and g be functions. If their domains overlap, then the overlap is the domain
of the following functions, with one exception (*):
( )( ) ( ) ( )
f g , where ( f g ) ( x ) = f ( x )
(Section 2.4: Limits and Infinity II) 2.4.4
PART C: RATIONAL FUNCTIONS
Example 2 (Contrast with Example 1)
Evaluate lim+
x
0
1
1
1
, lim 2 , and lim 2 .
x 0x
x2 x 0 x
Solution
Because x 2 > 0 for all nonzero values of x, all three give the
1
.
Limit Form
(Chapter 1: Review) 1.20
The term odd function may have come from the following fact:
()
If f x = x n , where n is an odd integer, then f is an odd function.
These are the functions for: , x 3 , x 1 , x1 , x 3 , x 5 , .
The reciprocal of a nonzero odd fun
(Section 2.4: Limits and Infinity II) 2.4.3
1
1
AND
0+
0
PART B : THE LIMIT FORMS
Example 1 showed us that lim+
x
0
1
=
x
, and lim
x
0
1
=
x
.
More generally, it is true that, for functions N and D,
()
D ( x)
()
Nx
if N x
()
1, and D x
0+ .
This is true
(Chapter 1: Review) 1.19
A function f is odd
f
( x) = f ( x)
x
()
Dom f
()
The graph of y = f x is symmetric
about the origin.
Example
()
If f x = x 3 , then f is odd, because
f
x
R,
( x) = ( x)
3
= x3
=
()
fx
()
The snake graph of f x = x 3 below is symm
(Section 2.4: Limits and Infinity II) 2.4.2
()
Similarly, lim f x =
x
0
.
()
In words: the limit of f x as x approaches 0 from the left is
negative infinity. That is, as x approaches 0 from lesser numbers, the
function values f x (generally) decrease with
(Chapter 1: Review) 1.18
The term even function may have come from the following fact:
()
If f x = x n , where n is an even integer, then f is an even function.
These are the functions for: , x 4 , x 2 , x 0 , x 2 , x 4 , .
The graph for the x 2 function
(Chapter 1: Review) 1.12
Example 6
()
If f x =
x+3
, find Dom f , the domain of f .
x 10
()
Solution
Because of the square root radical, we require:
x+3 0
x
3
Because we forbid zero denominators, we also require:
x 10 0
x 10
The domain of f
in set-build
(Section 2.3: Limits and Infinity I) 2.3.27
PART J: A WORD PROBLEM
Example 21 (Salt water mixtures)
A deep well contains 10 gallons of pure water at noon. Starting at noon, a
salt-water mixture is poured into the well at the rate of 2 gallons per
minute.
(Section 2.3: Limits and Infinity I) 2.3.21
How can we find an equation for the slant asymptote (SA)?
We can use the Long Division technique from Precalculus to
5 + 3x 2 + 6 x 3
reexpress f x =
as:
1 + 3x 2
()
()
f x = 2x + 1 +
polynomial
part, p( x )
= 2
(Chapter 1: Review) 1.05
The implied domain of an algebraic function consists of all real numbers except
those that:
1) lead to zero denominators Think:
0
, or
(
2) lead to negative radicands of even roots Think:
even
).
As we study more types of function
(Section 2.3: Limits and Infinity I) 2.3.20
Example 15
5 + 3x 2 + 6 x 3
.
Evaluate lim f x and lim f x , where f x =
x
x
1 + 3x 2
()
()
()
Solution 1 (A Rigorous Solution): Left to the reader! Refer to Example 13.
Solution 2 (The Short Cut: Dominant Term
(Chapter 1: Review) 1.04
Heres a Venn diagram for standard symbolic mathematical expressions:
PART E: DOMAIN AND RANGE
()
The domain of a function f , abbreviated Dom f , is the set of all legal inputs.
The range of f is then the set of all resulting outp
(Section 2.3: Limits and Infinity I) 2.3.19
Example 14
()
()
x2 3
Evaluate lim f x , where f x = 3
.
x
x + 4x2 + 1
Solution 1 (A Rigorous Solution): Left to the reader! Refer to Example 13.
Solution 2 (The Short Cut: Dominant Term Substitution)
()
x2 3
li
(Chapter 1: Review) 1.03
PART C: RATIONAL FUNCTIONS
A rational expression in x can be expressed in the form:
Examples:
1
5x3 1
,2
, x7 + x
x x + 7x
2
polynomial in x
.
nonzero polynomial in x
which equals
x7 + x
1
Observe in the second example that irrati
(Section 2.3: Limits and Infinity I) 2.3.18
Solution Method 2 (The Short Cut: Dominant Term Substitution)
()
()
Again, N x = 4 x 3 + x 1, and D x = 5x 3
()
Then, f x =
( ).
D ( x)
2x .
Nx
()
In our limit analysis, we may replace N x with its dominant term