in previous sections we have been using calculators and graphs to guess the values of limits....
TRANSCRIPT
In previous sections we have been using calculators and graphs to guess the values
of limits.
Sometimes, these methods do not work!
In this section we will use properties of limits, called limit laws, to calculate limits.
Calculating Limits Using the Limit Laws
Limit Laws:Suppose that c is a constant and the limits
and
exist. Then
1.
The limit of a sum is the sum of the limits.
Calculating Limits Using the Limit Laws
)(lim xfax
)(lim xgax
)(lim)(lim )()(lim xgax
xfax
xgxfax
Limit Laws:
2.
The limit of a difference is the difference of the limits.
Calculating Limits Using the Limit Laws
)(lim- )(lim )()(lim xgax
xfax
xgxfax
Limit Laws:
3.
The limit of a constant times a function is the constant times the limit of the function.
Calculating Limits Using the Limit Laws
)(lim )(lim xfax
cxcfax
Evaluate the limit and justify each step.
Example 1:
Note: If we let f(x)=2x2-3x+4, then f(5)=39.
Calculating Limits Using the Limit Laws
)4 3 - 22(5
lim
xxx
Limit Laws:
4.
The limit of a product is the product of the limits.
Calculating Limits Using the Limit Laws
)()(lim )()(lim xgxfax
xgxfax
Limit Laws:
5.
The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).
Calculating Limits Using the Limit Laws
0 )(lim if )(lim
)(lim
)()(lim
xgaxxg
ax
xfax
xgxf
ax
Evaluate the limit and justify your answer.
Example 2:
Note: As in example 1, if we let f(x)=(x3+2x2-1)/(5-3x), thenf(-2)=-1/11
Calculating Limits Using the Limit Laws
xxx
x 351223
2lim
Direct Substitution Property:
If f is a polynomial or a rational function and a is in the domain of f, then
Calculating Limits Using the Limit Laws
)()(lim afxfax
Let’s try example 1 on page 111.
Calculating Limits Using the Limit Laws
Example 4: Find
Calculating Limits Using the Limit Laws
112
1lim
xx
x
Limit Laws cont:
6. where n is a positive integer
Calculating Limits Using the Limit Laws
nxf
ax
nxf
ax
)(lim )(lim
Special Limits:
7.
8.
Calculating Limits Using the Limit Laws
ccax
lim
axax
lim
More Special Limits:
9. where n is a positive integer
10.
Calculating Limits Using the Limit Laws
nanxax
lim
]0a that assume weeven, isn [if
integer positive a isn where lim
n an xax
More Limit Laws:
11.
where n is a positive integer[If n is even, we assume that ]
Calculating Limits Using the Limit Laws
n xfax
n xfax
)(lim lim
0 )(lim
xfax
Example 5:
Find
Remember: The limit is not necessarily what the function is equal to; it is what the function approaches when x gets close to a
Calculating Limits Using the Limit Laws
1 xif 1 xif 1)( where)(
1lim
xxgxg
x
Example 6:
Evaluate:
Sometimes we may have to use algebra to take limits.
Calculating Limits Using the Limit Laws
hh
h
92)3(
0lim
Example 7:
Find
Here we have to rationalize the numerator in order to find the limit.
Calculating Limits Using the Limit Laws
2392
0lim
t
t
t
Theorem:
Some limits are best calculated by first finding the left- and right- hand limits. This theorem is a reminder of what we discovered in previous sections.
Calculating Limits Using the Limit Laws
)(lim)(-
limifonly and if )(lim xfax
Lxfax
Lxfax
Example 8:
Show that
Calculating Limits Using the Limit Laws
00
lim
xx
Example 9:
Prove that does not exist.
Calculating Limits Using the Limit Laws
x
x
x 0lim
Example 10:
Show that does not exist.
Calculating Limits Using the Limit Laws
xx 3lim
Theorem: If f(x) < g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then
Calculating Limits Using the Limit Laws
)(lim)(lim xgax
xfax
The Squeeze Theorem:
If f(x) < g(x) < h(x) when x is near a (except possibly at a) and
then
Calculating Limits Using the Limit Laws
Lxhax
xfax
)(lim)(lim
Lxgax
)(lim
Example 11:
By using the Squeeze Theorem show that
Calculating Limits Using the Limit Laws
01sin20
lim x
xx