Definition of Limit, Properties of Limits

Definition of Limit,Properties of

LimitsSection 2.1a

Let’s start with an exploration…What are the values of the function given below asx approaches 0???

sin xf xx

First, graph the function in thewindow by 2 ,2 1,2

Now, look at a table, withTblStart = –0.3 and Tbl = 0.1

–.3 .98507–.2 .99335–.1 .998330 ERROR.1 .99833.2 .99335.3 .99507

Let’s start with an exploration…What are the values of the function given below asx approaches 0???

sin xf xx

–.3 .98507–.2 .99335–.1 .998330 ERROR.1 .99833.2 .99335.3 .99507

What do these steps suggest?

Note: We cannot simply substitute x = 0into the function, because we’d be dividingby zero…………we need another method…

Definition: LimitLet c and L be real numbers. The function f has limit Las x approaches c if, given any positive number ,there is a positive number such that for all x,

0 x c f x L

We write limx cf x L

which is read, “the limit of f of x as x approaches c equals L.”

The notation means that the values of f(x) of the function fapproach or equal L as the values of x approach (but do notequal ) the number c…

Definition: LimitLet c and L be real numbers. The function f has limit Las x approaches c if, given any positive number ,there is a positive number such that for all x,

0 x c f x L

We write limx cf x L

As suggested in our opening exploration:

0

sinlimx

xx

1

Important note: The existence of a limit as x c never dependson how the function may or may not be defined at c.

2 11

xf xx

, 1

1, 1

f x xg x

x

1h x x

1 1 1

lim lim lim 2x x xf x g x h x

Definition: LimitLet c and L be real numbers. The function f has limit Las x approaches c if, given any positive number ,there is a positive number such that for all x,

0 x c f x L

We write limx cf x L

Find each of the following limits:

limx c

k

k limx c

x

c

Properties of LimitsIf L, M, c, and k are real numbers and

limx cf x L

and lim

x cg x M

, then

1. Sum Rule – The limit of the sum of two functions is the sum oftheir limits: lim

x cf x g x L M

2. Difference Rule – The limit of the difference of two functionsis the difference of their limits:

limx c

f x g x L M

Properties of LimitsIf L, M, c, and k are real numbers and

limx cf x L

and lim

x cg x M

, then

3. Product Rule – The limit of a product of two functions is theproduct of their limits:

limx c

f x g x L M

4. Constant Multiple Rule – The limit of a constant times afunction is the constant times the limit of the function:

limx c

k f x k L

Properties of LimitsIf L, M, c, and k are real numbers and

limx cf x L

and lim

x cg x M

, then

5. Quotient Rule – The limit of a quotient of two functions is thequotient of their limits, provided the limit of the denominator isnot zero:

lim , 0x c

f x L Mg x M

Properties of LimitsIf L, M, c, and k are real numbers and

limx cf x L

and lim

x cg x M

, then

6. Power Rule – If r and s are integers, , then

limr s r s

x cf x L

0s

provided that is a real number.r sLThe limit of a rational power of a function is that power of thelimit of the function, provided the latter is a real number.

Guided PracticeFind each of the following limits.

(a) 3 2lim 4 3x c

x x

3 2lim lim4 lim3x c x c x cx x

3 24 3c c

(b)4 2

2

1lim5x c

x xx

4 2

2

lim 1

lim 5x c

x c

x x

x

4 2

2

lim lim lim1

lim lim5x c x c x c

x c x c

x x

x

4 2

2

15

c cc

Noticeanything?

Theorem:Polynomial and Rational Functions

1. If 11 0

n nn nf x a x a x a

11 0lim n n

n nx cf x f c a c a c a

is anypolynomial function and c is any real number, then

2. If f x

lim ,x c

f x f cg x g c

are polynomials and c is any realnumber, then

and g x

0g c provided that

Guided PracticeFind each of the following limits.

(a) 2

3lim 2x

x x

23 2 3 9

(b)2

2

2 4lim2x

x xx

22 2 2 42 2

12 34

Guided PracticeExplain why you cannot use substitution to determine the givenlimits. Find the limit if it exists.

20

1limx x

21 xCannot use substitution b/c the expressionis not defined at x = 0.

21 xSince becomes arbitrarily large as x approaches0 from either side, there is no (finite) limit.

Can we support this reasoning graphically???

Guided PracticeExplain why you cannot use substitution to determine the givenlimits. Find the limit if it exists.

20

4 16limx

xx

Cannot use substitution b/c theexpression is not defined at x = 0.

A key rule for evaluating limits:If substitution will not work directly, use

algebra techniques to re-write the expressionso that substitution will work!!!

Guided PracticeExplain why you cannot use substitution to determine the givenlimits. Find the limit if it exists.

20

4 16limx

xx

Cannot use substitution b/c theexpression is not defined at x = 0.

2

0

16 8 16limx

x xx

2

0

8limx

x xx

0

8limx

x xx

0

lim 8x

x

8 0 8 Support graphically???

Guided PracticeDetermine the given limits algebraically. Support graphically.

2

22

3 2lim4t

t tt

2

1 2lim

2 2t

t tt t

2

1lim

2t

tt

2 1 12 2 4

Guided PracticeDetermine the given limits algebraically. Support graphically.

0

1 12 2lim

x

xx

0

2 22 2

limx

xx

x

0

1lim4 2x

xx x

0

1lim4 2x x

14

0

4 2limx

xx

x

Guided PracticeDetermine the given limits algebraically. Support graphically.

0

sin 2limx

xx 0

sin 22lim2x

xx

2 1 2

0

sinlimx

x xx

0

sinlim 1x

xx

0 0

sinlim1 limx x

xx

1 1 2