CalculusCalculusLimits Lesson 2Limits Lesson 2

Bell ActivityA. Use your calculator graph to find:

)23(lim 2 xx1.

)52(lim 3 xx2.

3. )2

4(lim 4x

x

4.)2()4(lim

2

2

xx

x

B. Without Calculator, FindB. Without Calculator, Find1. f(2) if f(x) = 3x – 22. f(-3) if f(x) = 2x + 53. f(4) if f(x) = 4 – x/24. f(2) if f(x) =

)2()4( 2

xx

Notice: In #4, both the numerator and denominator are 0. This is called an indeterminate form and thus cannot be evaluated.

The limit may be estimated or found by creating a table of values very close to the approaching x value.

)2()23(lim

2

2

xxx

x

X 1.75 1.9 1.99 2 2.001 2.01 2.1

Y .75 .9 .99 1.001 1.01 1.1

=1

X 2.8 2.9 2.99 3 3.001 3.01 3.1

Y

*** )52(lim 3 xx

What do you think allows this function to have the limit as x approaches 3 and f(3) to be the same whereas the first problem does not?

It would seem from this example that at times a limit of a function as x approaches a particular # can simply be obtained by finding f(that #).

=1

.6 .8 .98 1 1.002 1.02 1.2

)52(lim 3 xx

223lim

2

2

xxx

x

= 1

=1

Which of these can be substituted directly to find the limit?

11lim

11lim

)2(lim

2

1

2

2

23

xxxx

x

x

x

x1.

2.

3.

Even though # 3’s limit can’t be found by substituting directly we can still get the limit from a graph or a table.

=11

=3

=2

Let’s go back to a problem where the substitution would not work to find the limit. Perhaps we could modify the problem before we substituted.

11lim

2

1

xx

x

1)1)(1(lim 1

x

xxx

)1(lim 1 xx =2

Modify these and find the limit by substituting:

96lim 2

2

3

xxx

x 11lim

3

1

xx

x

Let’s verify these with the calculator graph.

Find the limit:

11lim

2

1

xx

x

Let’s verify with the calculator graph.

Find the limit:

Let’s verify with the calculator graph.

There are other times when we cannot find the limit by either substituting or factoring.

xx

x11lim 0

In this case, Rationalize the Numerator

And, of course, there are problems which merely need to be simplified before substituting.

xxxx

x

33

0)(lim

Find the limit:

Let’s verify with the calculator graph.

Find the limit:

Let’s verify with the calculator graph.

Find the limit:

Let’s verify with the calculator graph.

Slide 2- 17

Properties ofProperties of Limits Limits

If , , , and are real numbers and lim and lim , then

1. : lim

The limit of the sum of two functions is the sum of their limits.

2. : lim

The limit

x c x c

x c

x c

L M c kf x L g x M

Sum Rule f x g x L M

DifferenceRule f x g x L M

of the difference of two functions is the differenceof their limits.

Slide 2- 18

Properties ofProperties of Limits continuedLimits continued

3. lim

The limit of the product of two functions is the product of their limits.

4. lim

The limit of a constant times a function is the constant times the limitof the function.

5.

x c

x c

f x g x L M

k f x k L

Quot

: lim , 0

The limit of the quotient of two functions is the quotientof their limits, provided the limit of the denominator is not zero.

x c

f x Lient Rule Mg x M

Product Rule:

Constant Multiple Rule:

Properties ofProperties of Limits continuedLimits continued

6. : If and are integers, 0, then

lim

provided that is a real number.The limit of a rational power of a function is that power of the limit of the function, provided the latte

rrss

x c

rs

Power Rule r s s

f x L

L

r is a real number.

Other properties of limits:lim

limx c

x c

k k

x c

Example Properties ofExample Properties of LimitsLimits

3

Use any of the properties of limits to find

lim 3 2 9x c

x x

3 3

3

sum and difference rules

product and multiple rules

lim 3 2 9 lim3 lim2 lim9

3 2 9x c x c x c x c

x x x x

c c

AssignmentAssignment

Text p. 67, # 1 – 43 odds