l’hopital’s rule
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L’Hopital’s Rule
Objective: To use L’Hopital’s Rule to solve problems involving limits.
L’Hopital’s Rule
Forms of Type 0/0
• We looked at several limits in chapter 2 that had the form 0/0. Some examples were:
1sin
lim0
x
xx
21
1lim
2
1
x
xx
0cos1
lim0
x
xx
Forms of Type 0/0
• We looked at several limits in chapter 2 that had the form 0/0. Some examples were:
• The first limit was solved algebraically and the next two were solved using geometric methods. There are other types that cannot be solved by either of these methods. We need to develop a more general method to solve such limits.
1sin
lim0
x
xx
21
1lim
2
1
x
xx
0cos1
lim0
x
xx
Forms of Type 0/0
• Lets look at a general example. 00
)(
)(lim xg
xfax
Forms of Type 0/0
• Lets look at a general example.
• We will assume that f / and g / are continuous at x = a and g / is not zero. Since f and g can be closely approximated by their local linear approximation near a, we can say
00
)(
)(lim xg
xfax
))(()(
))(()(lim
)(
)(lim
/
/
axagag
axafaf
xg
xfaxax
Forms of Type 0/0
• Lets look at a general example.
• We know that:
• So we can now say
00
)(
)(lim xg
xfax
0)(lim)(
xfafax
0)(lim)(
xgagax
))((
))((
))(()(
))(()(lim
)(
)(lim
/
/
/
/
axag
axaf
axagag
axafaf
xg
xfaxax
Forms of Type 0/0
• Lets look at a general example.
• This will now become
• This is know as L’Hopital’s Rule.
00
)(
)(lim xg
xfax
)(
)(
)(
)(
)(
)(lim
/
/
/
/
xg
xf
ag
af
xg
xfax
Theorem 4.4.1
• L’Hopital’s Rule: Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that
and
• If exists or if this limit is then
0)(lim
xfax
.0)(lim
xgax
)](/)([lim // xgxfax
)(
)(lim
)(
)(lim
/
/
xg
xf
xg
xfaxax
Example 1
• Find the following limit. 2
4lim
2
2
x
xx
Example 1
• Find the following limit.
• In chapter 2 we would factor, cancel, and evaluate.
2
4lim
2
2
x
xx
42)2(
)2)(2(
2
4lim
2
2
x
x
xx
x
xx
Example 1
• Find the following limit.
• Now, we can use L’Hopital’s Rule to solve this limit.
2
4lim
2
2
x
xx
4)2(21
2
2
4lim
2
2
x
x
xx
Example 2
• Find the following limit. x
xx
2sinlim
0
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
x
xx
2sinlim
0
0
0
0
0sin2sinlim
0
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit.
x
xx
2sinlim
0
21
)1(2
1
2cos22sinlim
0
x
x
xx
Example 2
• Find the following limit. x
xx cos
sin1lim
2/
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
x
xx cos
sin1lim
2/
0
0
0
11
cos
sin1lim
2/
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit.
x
xx cos
sin1lim
2/
01
0
sin
cos
cos
sin1lim
2/
x
x
x
xx
Example 2
• Find the following limit. 30
1lim
x
ex
x
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
30
1lim
x
ex
x
0
0
0
111lim
30
x
ex
x
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit.
• This is an asymptote. Do sign analysis to find the answer. ___+___|___+___
0
30
1lim
x
ex
x
0
1
3
1lim
230 x
e
x
e xx
x
Example 2
• Find the following limit. 20
tanlim
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
20
tanlim
x
xx
0
0tanlim
20
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.• Next we use L’Hopital’s Rule to find the answer.
• Again, sign analysis. ___-___|___+___ 0
20
tanlim
x
xx
0
1
2
sectanlim
2
20 x
x
x
xx
Example 2
• Find the following limit. 20
cos1lim
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
20
cos1lim
x
xx
0
0
0
11cos1lim
20
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.• Next we use L’Hopital’s Rule to find the answer.
• We apply L’Hopital’s Rule again.
20
cos1lim
x
xx
0
0
2
sincos1lim
20
x
x
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.• Next we use L’Hopital’s Rule to find the answer.
• We apply L’Hopital’s Rule again.
20
cos1lim
x
xx
0
0
2
sincos1lim
20
x
x
x
xx
2
1
2
cos
2
sincos1lim
20
x
x
x
x
xx
Example 2
• Find the following limit. )/1sin(lim
3/4
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.
)/1sin(lim
3/4
x
xx
0
0
)/1sin(lim
3/4
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.• Next we use L’Hopital’s Rule to find the answer.
)/1sin(lim
3/4
x
xx
01
0
)/1cos(
)3/4(
)/1cos()/1(
)3/4(
)/1sin(lim
3/1
2
3/73/4
x
x
xx
x
x
xx
Example 2
• Find the following limit.
• First, we confirm that this of the form 0/0.• Next we use L’Hopital’s Rule to find the answer.
)/1sin(lim
3/4
x
xx
01
0
)/1cos(
)3/4(
)/1cos()/1(
)3/4(
)/1sin(lim
3/1
2
3/73/4
x
x
xx
x
x
xx
04
)/1cos(3
4
)/1cos()/1(
)3/4(
)/1sin(lim
3/12
3/73/4
xxxx
x
x
xx
Forms of Type
• Theorem 4.4.2: Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that
and
• If exists or if this limit is then
)(lim xgax
)(lim xfax
/
)](/)([lim // xgxfax
)(
)(lim
)(
)(lim
/
/
xg
xf
xg
xfaxax
Example 3
• Find the following limits. xx e
xlim
Example 3
• Find the following limits.
• First, we confirm the form of the equation.
xx e
xlim
xx e
xlim
Example 3
• Find the following limits.
• First, we confirm the form of the equation.• Next, apply L’Hopital’s Rule.
xx e
xlim
011
lim
xxx ee
x
Example 3
• Find the following limits.x
xx csc
lnlim0
Example 3
• Find the following limits.
• First, we confirm the form of the equation.
x
xx csc
lnlim0
x
xx csc
lnlim0
Example 3
• Find the following limits.
• First, we confirm the form of the equation.• Next, apply L’Hopital’s Rule.
• This is the same form as the original, and repeated applications of L’Hopital’s Rule will yield powers of 1/x in the numerator and expressions involving cscx and cotx in the denominator being of no help. We must try another method.
x
xx csc
lnlim0
xx
x
x
xx cotcsc
/1
csc
lnlim0
Example 3
• Find the following limits.
• We will rewrite the last equation as
x
xx csc
lnlim0
x
x
x
xx
x
x
xx
tansin
cotcsc
/1
csc
lnlim0
0)0)(1(tanlimsin
lim00
xx
xxx
Examine Exponential Growth
• Look at the following graphs. What these graphs show us is that ex grows more rapidly than any integer power of x.
0lim x
n
x e
x
n
x
x x
elim
Forms of Type
• There are several other types of limits that we will encounter. When we look at the form we can make it look like 0/0 or and use L’Hopital’s Rule.
0
0
/
Example 4
• Evaluate the following limit. xxx
lnlim0
Example 4
• Evaluate the following limit.
• As written, this is of the form
xxx
lnlim0
0lnlim0
xxx
Example 4
• Evaluate the following limit.
• As written, this is of the form
• Lets rewrite this equation as
0lnlim0
xxx
xxx
lnlim0
x
xx /1
lnlim0
Example 4
• Evaluate the following limit.
• As written, this is of the form
• Lets rewrite this equation as
• Applying L’Hopital’s Rule
0lnlim0
xxx
xxx
lnlim0
x
xx /1
lnlim0
0/1
/1
/1
lnlim
20
x
x
x
x
xx
Example 4
• Evaluate the following limit. xxx
2sec)tan1(lim4/
Example 4
• Evaluate the following limit.
• As written, this is of the form
02sec)tan1(lim4/
xxx
xxx
2sec)tan1(lim4/
Example 4
• Evaluate the following limit.
• As written, this is of the form
• Lets rewrite this equation as
02sec)tan1(lim4/
xxx
xxx
2sec)tan1(lim4/
x
xx 2cos
tan1lim
4/
Example 4
• Evaluate the following limit.
• As written, this is of the form
• Lets rewrite this equation as
• Applying L’Hopital’s Rule
02sec)tan1(lim4/
xxx
xxx
2sec)tan1(lim4/
x
xx 2cos
tan1lim
4/
12
2
2sin2
sec
2cos
tan1lim
2
4/
x
x
x
xx
Forms of Type
• There is another form we will look at. This is a limit of the form We will again try to manipulate this equation to match a form we like.
.
Example 5
• Find the following limit.
xxx sin
11lim0
Example 5
• Find the following limit.
• This is of the form
xxx sin
11lim0
Example 5
• Find the following limit.
• This is of the form
• Lets rewrite the equation as
xxx sin
11lim0
xx
xx
xxx sin
sin
sin
11lim0
Example 5
• Find the following limit.
• This is of the form
• Lets rewrite the equation as
• This is of the form 0/0, so we will use L’Hopital’s Rule.
xxx sin
11lim0
xx
xx
xxx sin
sin
sin
11lim0
02
0
cos2sin
sin
0
0
sincos
1cos
sin
sinlim0
xxx
x
xxx
x
xx
xxx
Forms of Type
• Limits of the form give rise to other indeterminate forms of the types:
)()(lim xgxf
1,,0 00
1,,0 00
Forms of Type
• Limits of the form give rise to other indeterminate forms of the types:
• Equations of this type can sometimes be evaluated by first introducing a dependent variable
and then finding the limit of lny.
)()(lim xgxf
1,,0 00
1,,0 00
)()( xgxfy
Example 6
• Show that ex x
x
/1
0)1(lim
Example 6
• Show that
• This is of the form so we will introduce a dependent variable and solve.
ex x
x
/1
0)1(lim
1
Example 6
• Show that
• This is of the form so we will introduce a dependent variable and solve.
ex x
x
/1
0)1(lim
1
xxy /1)1(
xxy /1)1ln(ln
)1ln()/1(ln xxy
x
xy
)1ln(ln
Example 6
• Show that ex x
x
/1
0)1(lim
0
0)1ln(limlnlim
00
x
xy
xx
x
xy
)1ln(ln
Example 6
• Show that ex x
x
/1
0)1(lim
0
0)1ln(limlnlim
00
x
xy
xx
x
xy
)1ln(ln
11
1
1
)1/(1)1ln(lim
0
x
x
x
xx
Example 6
• Show that
• This implies that as .
ex x
x
/1
0)1(lim
11
1
1
)1/(1)1ln(lim
0
x
x
x
xx
1lnlim0
yx
ey 0x
Homework
• Section 3.6• Page 226-227• 1-29 odd
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