limits of a function
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Limits of a Function
Review
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The Limit of a Function
If f(x) gets closer and closer to a number L as
x gets closer and closer to c from both sides,
then L is the limit of f(x) as x approaches c .
The behavior is expressed by writing
lim f ( x ) = L x c
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x
y
f ( x )
f ( x )
L
x c x
If lim f ( x ) = L, the height of the graph of f approaches Las x approaches c.
x c
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Limits at Infinity
As the values of the function f ( x ) approach the
number L as x increases without bound, we write
lim f ( x ) = L
x +
Similarly, we write
lim f ( x ) = M x -
When the functional values f ( x ) approach the
number M as x decreases without bound.
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Reciprocal Power Rule
If A and k are constants with k > 0 and xk
is defined for all x, then
lim A x kx +
=0 lim A x kx -
=0
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Consider the function
Evaluate the limit of f ( x )as x + .
2
2
21)(
x x
x x f
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Evaluate
253
132 2
lim
x x
x x
x
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Procedure for Evaluating Limits at
Infinity
Step 1. Divide each term in f ( x ) by the highest
power x k that appears in the denominator
polynomial
Step 2. Compute for the limit of f( x ) as x
approaches + and as x approaches -
using the algebraic properties of limits and
the reciprocal power rule.
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Exercises
A business manager determines that t months
after production begins on a new product, the
number of units produced will be P thousand,
where
2
2
1
56)(
t
t t t P
what happens to production in the long run?
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y
x
A continuous graph
y
xa b
A graph with “holes” or “gaps” is not continuous
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c
f ( c ) is not defined
y
x
y
xc
)()(lim c f x f c x
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JumpDiscontinuity RemovableDiscontinuity InfiniteDiscontinuity
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One-Sided LimitsThe function f has the right-hand limit L as x approaches
c, written
If the value of f ( x) can be made as close to the number L
as we can by taking x sufficiently close to (but not equalto) c and to the right of c.
L x f c x
)(lim
The function f has the left-hand limit M as x approaches a,
written
If the value of f ( x) can be made as close to the number M as
we can by taking x sufficiently close to (but not equal to) c
and to the left of c.
M x f c x
)(
lim
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One Sided Continuity
A function f is continuous from the right
at a number c if
and f is continuous from the left at a if
)()(lim c f x f c x
)()(lim c f x f c x
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Existence of Limits
The limit L of a function f ( x) at x = c exists if
and only if L is a real number and
c x approaching c
from the left.
x approaching c
from the right.
x
L x f x f c xc x
)()( limlim
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Continuity
The function, f is continuous at x = c if thefollowing three conditions are satisfied:
)()(3.
exists)(.2
definedis)(.1
lim
lim
c f x f
x f
c f
c x
c x
Polynomials, including constants, linear functions and
quadratic functions, are continuous as every value of
x.
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Fur ther Explanation: Continuity
A function is continuous at a number c if
That is, to prove a function is continuous at c, you must show
the following three things.
1. f (c) is defined – Evaluate f (c).
2. exists –
Find it.
One Sided Limits Needed?
3. – Show This from (1) and (2).)()(lim c f x f c x
)(lim x f c x
)()(lim c f x f c x
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Defini tion: Continui ty On An I nterval
A function f is continuous on an interval
if it is continuous at every number in the
interval. (If f is defined on one side of anendpoint of the interval, we understand
continuous at the endpoints to mean
continuous from the right or continuous from the left ).
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Continuity on an Interval
A function f ( x ) is said to be continuous on an openinterval a < x < b if it is continuous at each point x =c in that interval.
Moreover, f is continuous on the closed interval a < x < b and
)()(lim a f x f a x
)()(lim b f x f b x
and
In other words, continuity on an interval means that the graph
of f is “one piece” throughout the interval.
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1. f + g 2. f – g
3. kf
4. fg5. f / g if g (c) 0
If f and g are continuous at a and c is a constant,then the following functions are also continuous
at c:
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(a) Any polynomial is continuous
everywhere; that is, it is continuous on = (-∞,+ ).
(b) Any rational function is continuous
whenever it is defined; that is, it iscontinuous on its domain.
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Remember
Any of the following types of functions are
continuous at every number in their domain:• Polynomials
• Rational Functions
• Root Functions
• Exponential Functions
• Logarithmic Functions
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If f is continuous at b and , then
. In other words,
lim ( ) x a
g x b
)())((lim b f x g f a x
))(lim())((lim x g f x g f a xa x
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If g is continuous at c and f is continuous at g (c),
then the composite function f ( g ( x)) is continuous
at c.
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Exercise
Discuss the continuity of the function
3
2
)(
x
x
x f
on the open interval -2 < x < 3 and on the
closed interval -2 < x < 3.
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Sample Problem
The graph below shows the amount of gasoline in thetank of Mark’s car over a 30-day period. When is thegraph discontinuous? What do you think happens atthese times?
10
10 25 3050 15 20
5
t (days )
Q(liters)
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Exercise
Find the values of A such that the function
f ( x ) will be continuous for all x .
Ax - 3 if x < 2
f ( x ) =3 – x + 2 x if x > 2
2
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The I ntermediate Value Property
Suppose that f is continuous on the closed interval [a, b] and let L be
any number between f (a) and f (b). Then there exists a number c in (a,
b) such that f (c) = L.
a
f
b
f (a)
f (b)
c
f (c)=L
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The Intermediate Value Property
If the function, f is continuous on a closed
interval [a,b] and f (a) < L < f (b) , then
there is at least one number c in [a,b] such
that f (c) = L.
Further if f (a) and f (b) have opposite
signs, then there is at least one solution ofthe equation f ( x) = 0 in the open interval
(a,b).
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Example
Use the Intermediate Value Property to show that
there is a root of the given equation in the
specified interval.
)2,1(,1
112
x x x
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Sample Problem
A business manager determines that when x % ofher company’s plant capacity is being used, thetotal cost of operation is C hundred thousandpesos, where
96068
3206368)(
2
2
x x
x x xC
a. Find C (0) and C (100).
b. Explain why the result of part (a) cannot be usedalong with the intermediate value property to showthat the cost of operation is exactly P700,000 whena certain percentage of plant capacity is being
used.
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Answers
a. C(0) = 0.3333(approx);
C(100) = 7.179 (approx)
b. C ( x ) is not continuous on the interval
0< x <100 because C (80) is not defined.