Calculus Limits Lesson 2. Bell Activity A. Use your calculator graph to find: 1. 2. 3. 4.
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Transcript of Calculus Limits Lesson 2. Bell Activity A. Use your calculator graph to find: 1. 2. 3. 4.
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