limit and continuity.pdf

7/27/2019 Limit and Continuity.pdf

http://slidepdf.com/reader/full/limit-and-continuitypdf 1/13

Math231 Class Notes Unit 2

Unit 2. Limits and Continuity

Definition: A function is continuous at a point if:a) a function exists at this point ( is defined for a given value of x)b) any small change in x produces only small changes in f(x).

Graphically in means that there is no breaks, jumps, asymptotes at thispoint.

A function is said continuous over the interval if it is continuous at eachpoint on the interval.

Polynomials are continuous everywhere.

For example y = x2 -1(parabola shifted 1 unit down)

is defined everywhere, x cantake on any value.

There are no breaks, it’ssmooth:

when x gradually changes,y changes also gradually.




Types of Discontinuity

1. Asymptotic ( infinite)

This function is not defined at x=2.When x approaches 2, thefunction increases infinitely.The vertical line x = 2 is called anasymptote. The graph cannottouch it or cross it.

2. “Hole”

)2(

)2)(3(

x

x x y

There is one undefined pointat x = 2 on otherwise linear graph y = x +3

it’s shown as an hollow circle – a hole

3. “Jump” Piece-wise function

16

13 2

x for

x for x y

The function exists for allvalues of x(defined

everywhere)But the left portion of the graphis a parabola y = 3x2, f(1) = 3while the right portion – horizontal line y = 6.

2

y = x + 3




Small change in x when close to 1 produces a sudden jump 3 units up.




More examples of discontinuity (or not)

1.

24

232

x for

x for x y

“Jump”

2.

112

142

x for x

x for x y

Discontinuity “jump” at x = 1




3. Piece-wise functions are not necessarily discontinuous.

If both functions produce thesame value on the border of the interval of existence,there will be no break in thegraph.

This function is continuous.

4.

)1(

)1)(2(

1

232

x

x x

x

x x y

Discontinuity “hole” at x = 1

5.

)1)(2(

)1(

x x

x y

For all points other than x = -1the bracket (x +1) can becancelled and the function turns

into2

1

x

y that has an

asymptotic discontinuity at x = -2.

1

2

2




At x = -1 function is not defined, therefore - discontinuity “hole”




Limits as x approaches a number

The symbol → means “approaches to”

Writing x→a means “ x is as close to a as you wish, but never x = a

We are interested in the function behaviour in the vicinity of a certainvalue of x.Definition: The limit of a function is that value which the function

approaches as x approaches given value a.

L is a limit.

Example 1. y = 2x + 1 x→2

It’s obvious from the table of values that the closer x is to 2, the closer y is to 5. Therefore:

5)12(lim2

x x

Generally, if the function is continuous, limit can be found by directsubstitution of the approaching x – value into the function:

)()(lim a f x f a x

It is possible that a function is NOT continuous at some point, f (a)cannot be found, but the limit still exists and can be determined.

Example 2.

34

3)(

2

x x

x x f

This function isdiscontinuous at x = 3since f(3) is not defined.Let’s examine the function From the table you can easily

behaviour when x→3 infer that the function tends to

reach value of 0.5 when x is getting closer and closer to 3 regardless from thesmaller values side or greater

x 1.9 1.99 1.999 … y 4.8 4.98 4.998 …

x 2.9 2.99 2.999 … y 0.52632 0.50251 0.50025 …

x 3.1 3.01 3.001 …

y 0.47619 0.49751 0.49976 …

L x f a x

)(lim

5.0)(lim3

x f x




values sideFor discontinuous functions we have to find limits (if they exist)without f(a). For some types of discontinuity limit does not exist:

Consider

1

1

x

y when x→1. Values of y will increase without any

tendency to any number. Therefore exist not does x x

1

1lim

1

One-sided Limits

Writing x→a+ means approaching a from the greater values, called

right-hand side limit.

Writing x→a- means approaching a from the smaller values, called

left-hand side limit.

x→a assumes both.

Existence of a limit

L x f a x

)(lim

requires that both one-sided limits exist and are equal:

L x f

L x f

a x

a x

)(lim

)(lim

The table in Ex.2 shows that both right-hand side limit and left-hand side

limit exist when x→3 and equal to 0.5. That’s why the general limit is0.5

Finding Limits for

)(

)()(

xQ

x P x f

Substitute x = a, get

the answer (works for

continuous functions

Substitute x = a, if the

answer is0

number

limit does not exist,

asymptotic discontinuity

Substitute x = a, if the answer

is0

0(called uncertainty)

limit may exist, (“hole”discontinuity). Use

factoring or rationalizing




Examples:Factoring

8)4(lim)4(

)4)(4(lim

0

0

4

16lim.1

44

2

4

x x

x x

x

x

x x x

2

1

1

1lim

)3)(1(

)3(lim

0

0

34

3lim.2

3323

x x x

x

x x

x

x x x

...0

1

1

1lim

)1)(1(

)1(lim

0

0

1

1lim.3

1121end or

x x x

x

x

x

x x x

Rationalizing

2

1

110

1

)11(lim

)11(

1)1(lim)

)11(

1)1(lim

)11(

)11)(11(lim

)()11(

)11(

0

011lim.4

0

0

22

00

0

x x

x

x x

x

x x

x

x x

x x

conjugatebymultiply x

x

x

x

x

x x x

x

41

0421

)42(lim

)42(

)4(4lim)

)42(

)4(2lim

)42(

)42)(4(2lim

)()42(

)42(

0

042lim.5

0

0

22

00

0

t t t

t t

t

t t

t

t t

t t

conjugatebymultiplyt

t

t

t

t

t xt

t




Simplifying Complex Fraction

4

1

)02(2

1

)2(2

lim)2(2

)2(2

lim

0

02

1

2

1

lim.6000

hh

h

h

h

h

h

hhhh

Expanding

404

4(lim

4lim

4)44(lim

0

04)2(lim.7

0

2

0

2

0

2

0

h

h

h

hh

h

hh

h

h

hhhh

Practice:

238lim.1

2

3

2

x x x

x

12

2lim.2

2

2

1

x x

x x

x

352

94lim.3

2

2

5.1

x x

x

x

9

26lim.4

9

x

x

x

x x

x

x

2

4

lim.52

4

62lim.6

2

2

2

x

x x

x




Limits as x approaches Infinity.

Infinity symbol is .Note: is not a real number , algebraic operations can’t be applied to it.

When x getting very big or very small, (x→±∞) we can figure out “end

behaviour” of the functions. Example: exponential function (see graph)

Basic limits for algebraic functions:

01

lim x x

Both results can be understood as

value small veryvaluebig very

1

Possible answers:

L x f x )(lim.1

Horizontal asymptote y = L

0)(lim.2

x f x

Horizontal asymptote y = 0

)(lim.3 x f x

No horizontal asymptote

)0(01

lim

r xr x

.)..(lim

0lim

end or e

e

x

x

x

x

y = L

y = 0




Tool: divide numerator and denominator over the highest power thatappears in the expression.

Examples:

22

4

12

84

lim12

84

lim12

84lim.1

2

2

22

2

22

2

2

2

x

x

x x

x x x

x

x

x

x x x

.)..(4

1

2

4lim

24

lim24

lim.23

3

2

33

3

2

3

end small very

x

x

x

x x x

x

x

x

x x x

02

0

72

521

lim72

52

lim72

52lim.3

3

32

33

3

333

2

3

2

x

x x x

x x

x x x

x

x

x

x

x x

x x x

Shortcuts:

1. If the highest powers in the numerator and denominator are thesame, the limit is the ratio of the coefficients of these powers.

2. If the power of the numerator is lower than that of denominator,the limit equals to zero.

3. If the power of the numerator is higher than that of denominator,the limit does not exist the function increases infinitely.

0

0

0

0

0

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