LIMITS OF FUNCTIONS. INFINITE LIMITS; VERTICAL AND HORIZONTAL ASYMPTOTES; SQUEEZE THEOREM OBJECTIVES: define infinite limits; illustrate the infinite.

LIMITSOF

FUNCTIONS

INFINITE LIMITS; VERTICAL AND HORIZONTAL ASYMPTOTES;

SQUEEZE THEOREMOBJECTIVES:•define infinite limits;•illustrate the infinite limits ; and•use the theorems to evaluate the limits of functions.•determine vertical and horizontal asymptotes•define squeeze theorem

DEFINITION: INFINITE LIMITS

Sometimes one-sided or two-sided limits fail to exist because the value of the function increase or decrease without bound. For example, consider the behavior of forvalues of x near 0. It is evident from the table and graph in Fig 1.1.15 that as x values are taken closer and closer to 0 from the right, the values ofare positive and increase without bound; and asx-values are taken closer and closer to 0 from the left, the values of are negative and decrease without bound.

x

1)x(f

x

1)x(f

x

1)x(f

In symbols, we write

x

1lim and

x

1lim

0x0x

Note:The symbols here are not real numbers; they simply describe particular ways in which the limits fail to exist. Thus it is incorrect to write .

and

0

Figure 1.1.15 (p. 74)

1.1.4 (p. 75) Infinite Limits (An Informal View)

Figure 1.2.2 (p. 84)

Figure 1.1.2 illustrate graphically the limits for rational functions of the form .

22 ax

1 ,

ax

1 ,

ax

1

EXAMPLE: Evaluate the following limits:

40x x

1 lim .1

40x x

1 lim .2

50x x

1 lim .4

0

1

x

1 lim

50x

0

1

x

1 lim

40x

0

1

x

1 lim

40x

0

1

x

1 lim

50x

50x x

1 lim .5

40x x

1 lim .3

50x x

1 lim .6

2x

x3 lim a. .7

2x

2x

x3 lim.b

2x

0

6

0

23

2-x

3x lim

-2x

2.028.12x

8.1 say ,left from2 to close x of value

take we means2x

0

6

0

23

2-x

3x lim

2x

1.021.22x

1.2 say ,right from2 to close x of value

take we means2x

2x

x3 lim .c

2x

)x(flimax

)x(flimax

)x(flimax

SUMMARY:

)x(Q

)x(R)x(f If

EXAMPLE

3

1

3x

2

3x

2 lim ,then

3

1

3x

2 lim and

3x

2 lim .1

3x

3x3x

c

c

nSubtractio/Addition:Note

c c

c c

:Note

11x

3x

1x

x2 lim ,then

11x

3x lim and

1x

x2 lim .2

1x

1x1x

3

1

4x

6x2

2x

x3 lim ,then

3

1

4x

6x2 lim and

2x

x3 lim .3

2x

2x2x

VERTICAL AND HORIZONTAL ASYMPTOTES

DEFINITION:

)x(flim.d

)x(flim.c

)x(flim.b

)x(flim.a

ax

ax

ax

ax

The line is a vertical asymptote of the graph of the function if at least one of the following statement is true:

x a y f x

x=a

0

)x(flimax

)x(flimax

The following figures illustrate the vertical asymptote . x a

x=a

0

x=a

0

x=a

)x(flimax

)x(flimax

The following figures illustrate the vertical asymptote . x a

0

DEFINITION:

b)x(flim or b)x(flimxx

The line is a horizontal asymptote of the graph of the function if either

by y f x

y=b

0

y=b

b)x(flimx

The following figures illustrate the horizontal asymptote

by

0

b)x(flimx

y=b

0

y=b

b)x(flim x

The following figures illustrate the horizontal asymptote by

0

b)x(flimx

Determine the horizontal and vertical asymptote of the function and sketch the graph. 3

2f x

x

a. Vertical Asymptote: Equate the denominator to zero to solve for the vertical asymptote.

2x02x

Evaluate the limit as x approaches 2

2

3 3 3lim

2 2 2 0x x

b. Horizontal Asymptote:Divide both the numerator and the denominator by the highest power of x to solve for the horizontal asymptote.

3 30

lim 02 2 1 01

x

xxx x

3 30

lim 02 2 1 01

x

xxx x

erceptintx no is there therefore

30 ;2x

30 ,0)xf( If

2

3

20

3xf ,0x If

:Intercepts

.asymptote horizontal a is 0 ,Thus

2

3,0

VA: x=2

HA:y=00

3

2f x

x

Determine the horizontal and vertical asymptote of the function and sketch the graph.

3x

1x2xf

a. Vertical Asymptote: b. Horizontal Asymptote:

3x03x

0

7

3x

1x2lim

3x

21

2

x3

xx

x1

xx2

limx

asymptote horizontal a is 2y asymptote vertical a is 3x

2

1x ;

3x

1x20 ,0)xf( If

3

1

30

10xf ,0x If

:Intercepts

HA:y=2

VA:x=3

o

3x

1x2xf

SQUEEZE THEOREM

LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE

The Squeeze Principle is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, etc.) are not effective. However, it requires that you be able to ``squeeze'' your problem in between two other ``simpler'' functions whose limits are easily computable and equal. The use of the Squeeze Principle requires accurate analysis, algebra skills, and careful use of inequalities. The method of squeezing is used to prove that f(x)→L as x→c by “trapping or squeezing” f between two functions, g and h, whose limits as x→c are known with certainty to be L.

SQUEEZE PRINCIPLE :

Lf(x)lim then

h(x)lim Lg(x)lim and

h(x)f(x)g(x) satisfy h and ,g ,f functions that Assume

ax

axax

Theorem 1.6.5 (p. 123)

Figure 1.6.3 (p. 123)

EXAMPLE:

x

cos4x-cos3x-2lim 4.

x5sin

x3sinlim .3

x

x2sinlim .2

x

xtanlim .1

limits. following the Evaluate

0x0x

0x0x

111

xcos

1lim

x

xsinlim

xcos

1

x

xsin lim

x

xtanlim .1

0x0x

0x0x

212 x2

x2sinlim2

2

2

x

x2sin lim

x

x2sinlim .2

0x

0x0x

SOLUTION:

5

3

15

13

x5x5sin

5

x3x3sin

3lim

xx5sin

xx3sin

limx5sin

x3sinlim .3

0x

0x0x

00403

4x

cos4x-1lim4

3x

cos3x-1lim3

x

cos4x-1lim

x

cos3x-1lim

x

x4cosx3cos11 lim

x

cos4x-cos3x-2lim 4.

0x0x

0x0x

0x

0x

3x2x

2xx lim .4

4x

x16 lim .3

4t

2t lim .2

x9

x4 lim .1

2

2

3x

2

4x

22t

2

2

3x

EXERCISES: Evaluate the following limits: