Limits of a Function

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Limits of a Function

Review



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



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



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.



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



Consider the function

Evaluate the limit of f ( x )as x + .

2

2

21)(

x x

x x f



Evaluate

253

132 2

lim

x x

x x

x



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.



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?



y

x

A continuous graph

y

xa b

A graph with “holes” or “gaps” is not continuous



c

f ( c ) is not defined

y

x

y

xc

)()(lim c f x f c x



JumpDiscontinuity RemovableDiscontinuity InfiniteDiscontinuity



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



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



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



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.



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



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 ).



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.



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:



(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.



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



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



If g is continuous at c and f is continuous at g (c),

then the composite function f ( g ( x)) is continuous

at c.



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.



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)



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



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



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).



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



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.



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.

Limits of a Function

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