l’hopital’s rule
DESCRIPTION
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:. Forms of Type 0/0. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/1.jpg)
L’Hopital’s Rule
Objective: To use L’Hopital’s Rule to solve problems involving limits.
![Page 2: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/2.jpg)
L’Hopital’s Rule
![Page 3: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/3.jpg)
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
![Page 4: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/4.jpg)
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
![Page 5: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/5.jpg)
Forms of Type 0/0
• Lets look at a general example. 00
)(
)(lim xg
xfax
![Page 6: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/6.jpg)
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
![Page 7: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/7.jpg)
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
![Page 8: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/8.jpg)
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
![Page 9: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/9.jpg)
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
![Page 10: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/10.jpg)
Example 1
• Find the following limit. 2
4lim
2
2
x
xx
![Page 11: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/11.jpg)
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
![Page 12: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/12.jpg)
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
![Page 13: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/13.jpg)
Example 2
• Find the following limit. x
xx
2sinlim
0
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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
![Page 15: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/15.jpg)
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
![Page 16: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/16.jpg)
Example 2
• Find the following limit. x
xx cos
sin1lim
2/
![Page 17: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/17.jpg)
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
![Page 18: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/18.jpg)
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
![Page 19: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/19.jpg)
Example 2
• Find the following limit. 30
1lim
x
ex
x
![Page 20: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/20.jpg)
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
![Page 21: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/21.jpg)
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
![Page 22: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/22.jpg)
Example 2
• Find the following limit. 20
tanlim
x
xx
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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
![Page 24: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/24.jpg)
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
![Page 25: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/25.jpg)
Example 2
• Find the following limit. 20
cos1lim
x
xx
![Page 26: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/26.jpg)
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
![Page 27: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/27.jpg)
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
![Page 28: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/28.jpg)
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
![Page 29: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/29.jpg)
Example 2
• Find the following limit. )/1sin(lim
3/4
x
xx
![Page 30: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/30.jpg)
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
![Page 31: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/31.jpg)
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
![Page 32: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/32.jpg)
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
![Page 33: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/33.jpg)
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
![Page 34: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/34.jpg)
Example 3
• Find the following limits. xx e
xlim
![Page 35: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/35.jpg)
Example 3
• Find the following limits.
• First, we confirm the form of the equation.
xx e
xlim
xx e
xlim
![Page 36: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/36.jpg)
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
![Page 37: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/37.jpg)
Example 3
• Find the following limits.x
xx csc
lnlim0
![Page 38: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/38.jpg)
Example 3
• Find the following limits.
• First, we confirm the form of the equation.
x
xx csc
lnlim0
x
xx csc
lnlim0
![Page 39: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/39.jpg)
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
![Page 40: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/40.jpg)
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
![Page 41: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/41.jpg)
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
![Page 42: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/42.jpg)
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
/
![Page 43: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/43.jpg)
Example 4
• Evaluate the following limit. xxx
lnlim0
![Page 44: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/44.jpg)
Example 4
• Evaluate the following limit.
• As written, this is of the form
xxx
lnlim0
0lnlim0
xxx
![Page 45: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/45.jpg)
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
![Page 46: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/46.jpg)
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
![Page 47: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/47.jpg)
Example 4
• Evaluate the following limit. xxx
2sec)tan1(lim4/
![Page 48: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/48.jpg)
Example 4
• Evaluate the following limit.
• As written, this is of the form
02sec)tan1(lim4/
xxx
xxx
2sec)tan1(lim4/
![Page 49: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/49.jpg)
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/
![Page 50: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/50.jpg)
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
![Page 51: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/51.jpg)
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.
.
![Page 52: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/52.jpg)
Example 5
• Find the following limit.
xxx sin
11lim0
![Page 53: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/53.jpg)
Example 5
• Find the following limit.
• This is of the form
xxx sin
11lim0
![Page 54: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/54.jpg)
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
![Page 55: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/55.jpg)
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
![Page 56: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/56.jpg)
Forms of Type
• Limits of the form give rise to other indeterminate forms of the types:
)()(lim xgxf
1,,0 00
1,,0 00
![Page 57: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/57.jpg)
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
![Page 58: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/58.jpg)
Example 6
• Show that ex x
x
/1
0)1(lim
![Page 59: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/59.jpg)
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
![Page 60: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/60.jpg)
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
![Page 61: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/61.jpg)
Example 6
• Show that ex x
x
/1
0)1(lim
0
0)1ln(limlnlim
00
x
xy
xx
x
xy
)1ln(ln
![Page 62: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/62.jpg)
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
![Page 63: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/63.jpg)
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
![Page 64: L’Hopital’s Rule](https://reader031.vdocuments.mx/reader031/viewer/2022032805/56813253550346895d98d67f/html5/thumbnails/64.jpg)
Homework
• Section 3.6• Page 226-227• 1-29 odd