Transcript 8 7 Hopitals Rule

8-7: L’Hôpital’s Rule
Objectives:
1. To apply L’Hôpital’s
Rule to evaluate limits
Assignment
• P.574-576: 11-51 eoo,
59, 60, 63, 64, 65-69
odd, 75, 91-94, 97
Warm Up 1
Evaluate
2𝑥 2 −2
lim
.
𝑥+1
𝑥→−1
When a limit
produces the
indeterminate form
0
, try factoring or
0
rationalizing.
Indeterminate form
2𝑥 2 − 2
0
lim
=
𝑥→−1 𝑥 + 1
0
2 𝑥+1 𝑥−1
lim
=
𝑥→−1
𝑥+1
lim 2 𝑥 − 1 = −4
𝑥→−1
Warm Up 2
Evaluate
3𝑥 2 −1
lim 2 .
𝑥→∞ 2𝑥 +1
When a limit
produces the
indeterminate form
∞
, try dividing by
∞
the highest power
in the denominator.
Indeterminate form
3𝑥 2 − 1
∞
lim 2
=
𝑥→∞ 2𝑥 + 1
∞
3𝑥 2 /𝑥 2 − 1/𝑥 2
lim 2 2
=
2
𝑥→∞ 2𝑥 /𝑥 + 1/𝑥
3
3 − 1/𝑥 2
lim
=
2
𝑥→∞ 2 + 1/𝑥
2
Warm Up 3
Evaluate
𝑒 2𝑥 −1
lim
.
𝑥→0 𝑥
0
𝑒 2𝑥 − 1
lim
=
𝑥→0
0
𝑥
𝑒𝑥 − 1 𝑒𝑥 + 1
lim
=
𝑥→0
𝑥
Unhelpful
Warm Up 3
Evaluate
𝑒 2𝑥 −1
lim
.
𝑥→0 𝑥
0
𝑒 2𝑥 − 1
≈2
lim
=
𝑥→0
0
𝑥
𝒙
−𝟎. 𝟏
−𝟎. 𝟎𝟏 −𝟎. 𝟎𝟎𝟏
𝟎
𝟎. 𝟎𝟎𝟏
𝟎. 𝟎𝟏
𝟎. 𝟏
𝑓 𝑥
1.813
1.980
−
2.002
2.020
2.214
1.998
Warm Up 3
Evaluate
𝑒 2𝑥 −1
lim
.
𝑥→0 𝑥
𝑦 = 𝑒 2𝑥 − 1
𝑦=𝑥
Using local linearity,
𝑦 = 𝑥 has slope of 1
at 𝑥 = 0, while
𝑦 = 𝑒 2𝑥 − 1 is trying
to look like 2 at 𝑥 = 0.
Thus we have
2/1 = 2 at the origin.
𝑚=1
𝑚=2
Objective 1
You will be able to apply
L’Hôpital’s Rule to evaluate
limits
Ratio of Slopes
As the previous
example
suggested,
when
𝑓 𝑎
𝑔 𝑎
the limit
0
= ,
0
𝑓 𝑎
of
𝑔 𝑎
is
equal to the limit
of the ratio of the
slopes at 𝑥 = 𝑎.
Ratio of Slopes
As the previous
example
suggested,
when
𝑓 𝑎
𝑔 𝑎
0
0
= .
At 𝑥 = 𝑎,
𝑓 𝑥 = 𝑓 𝑎 + 𝑓′ 𝑎 𝑥 − 𝑎 and
𝑔 𝑥 = 𝑔 𝑎 + 𝑔′ 𝑎 𝑥 − 𝑎
𝑓 𝑎
𝑔 𝑎
the limit
0
= ,
0
𝑓 𝑎
of
𝑔 𝑎
Assume
is
equal to the limit
of the ratio of the
slopes at 𝑥 = 𝑎.
𝑓 𝑥
𝑓 𝑎 + 𝑓′ 𝑎 𝑥 − 𝑎
= lim
𝑥→𝑎 𝑔 𝑥
𝑥→𝑎 𝑔 𝑎 + 𝑔′ 𝑎 𝑥 − 𝑎
lim
𝑓 𝑎 + 𝑓′ 𝑎 𝑥 − 𝑎
= lim
𝑥→𝑎 𝑔 𝑎 + 𝑔′ 𝑎 𝑥 − 𝑎
𝑓′ 𝑎 𝑥 − 𝑎
𝑓′ 𝑎
= lim
= lim
𝑥→𝑎 𝑔′ 𝑎 𝑥 − 𝑎
𝑥→𝑎 𝑔′ 𝑎
L’Hôpital’s Rule
If 𝑓 and 𝑔 are differentiable
on an open interval 𝑎, 𝑏
containing 𝑐 and 𝑔′ 𝑥 ≠ 0
for all 𝑥 in 𝑎, 𝑏 except
possibly at 𝑐, if the limit as
𝑥 → 𝑐 produces the
indeterminate form
0 ∞ −∞
∞
, , , or , then
0 ∞
∞
−∞
𝑓 𝑥
𝑓′ 𝑥
lim
= lim
𝑥→𝑐 𝑔 𝑥
𝑥→𝑐 𝑔′ 𝑥
Merci, M.
Bernoulli!
provided the limit on
the right exists or is
infinite.
Guillaume L’Hôpital, c. 1696
“L’Hôpital’s” Rule
If 𝑓 and 𝑔 are differentiable
on an open interval 𝑎, 𝑏
containing 𝑐 and 𝑔′ 𝑥 ≠ 0
for all 𝑥 in 𝑎, 𝑏 except
possibly at 𝑐, if the limit as
𝑥 → 𝑐 produces the
indeterminate form
0 ∞ −∞
∞
, , , or , then
0 ∞
∞
−∞
𝑓 𝑥
𝑓′ 𝑥
= lim
𝑥→𝑐 𝑔 𝑥
𝑥→𝑐 𝑔′ 𝑥
lim
Perhaps that
should be
“L’Hopital’s”
Rule.
provided the limit on
the right exists or is
infinite.
Johann Bernoulli, c. 1694
“L’Hôpital’s” Rule
If 𝑓 and 𝑔 are differentiable
on an open interval 𝑎, 𝑏
containing 𝑐 and 𝑔′ 𝑥 ≠ 0
for all 𝑥 in 𝑎, 𝑏 except
possibly at 𝑐, if the limit as
𝑥 → 𝑐 produces the
indeterminate form
0 ∞ −∞
∞
, , , or , then
0 ∞
∞
−∞
𝑓 𝑥
𝑓′ 𝑥
= lim
𝑥→𝑐 𝑔 𝑥
𝑥→𝑐 𝑔′ 𝑥
lim
provided the limit on
the right exists or is
infinite.
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒
L’H is
,
𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒
not
the Quotient Rule
for Derivatives.
“L’Hôpital’s” Rule
If 𝑓 and 𝑔 are differentiable
on an open interval 𝑎, 𝑏
containing 𝑐 and 𝑔′ 𝑥 ≠ 0
for all 𝑥 in 𝑎, 𝑏 except
possibly at 𝑐, if the limit as
𝑥 → 𝑐 produces the
indeterminate form
0 ∞ −∞
∞
, , , or , then
0 ∞
∞
−∞
𝑓 𝑥
𝑓′ 𝑥
= lim
𝑥→𝑐 𝑔 𝑥
𝑥→𝑐 𝑔′ 𝑥
lim
provided the limit on
the right exists or is
infinite.
L’H can be
extended to
limits at infinity
or one-sided
limits.
Exercise 1
Evaluate.
1.
2𝑥 2 −2
lim
𝑥→−1 𝑥+1
2.
3𝑥 2 −1
lim 2
𝑥→∞ 2𝑥 +1
3.
𝑒 2𝑥 −1
lim
𝑥→0 𝑥
Exercise 2
Evaluate
ln 𝑥
lim
.
𝑥→∞ 𝑥
Relatively,
𝑦 = ln 𝑥 is
growing much
slower than
𝑦 = 𝑥, so the limit
approaches 0.
𝑦=𝑥
𝑦 = ln 𝑥
Exercise 3
Evaluate
𝑥2
lim −𝑥.
𝑥→−∞ 𝑒
Relatively, 𝑦 = 𝑥 2
is growing much
slower than
𝑦 = 𝑒 −𝑥
(as 𝑥 → ∞), so
the limit
approaches 0.
𝑦 = 𝑥2
𝑦 = 𝑒 −𝑥
Exercise 4
Evaluate L’Hôpital’s original example illustrating
“his” rule.
3
lim
𝑥→𝑎
2𝑎3 𝑥 − 𝑥 4 − 𝑎 𝑎2 𝑥
4
𝑎 − 𝑎𝑥 3
Exercise 5
Evaluate lim 𝑒 −𝑥 𝑥.
𝑥→∞
L’H can’t be used.
lim 𝑒 −𝑥 𝑥 = 0 ∙ ∞
𝑥→∞
𝑥
∞
lim
=
𝑥→∞ 𝑒 𝑥
∞
Or can it?
Now it can.
Indeterminate Forms
Recall that in order to use L’H, the limit must
produce an indeterminate form of the form
0 ∞ −∞
, , ,
0 ∞ ∞
or
∞
−∞
However, there are a few more indeterminate
forms that we can trick into being one of
these forms.
Indeterminate Products
If lim 𝑓 𝑥 𝑔 𝑥
𝑥→𝑎
produces the
indeterminate form
0 ∙ ∞, then try
rewriting the
product as a
quotient.
𝑓
𝑔
𝑓∙𝑔=
=
1
1
𝑔
𝑓
Exercise 6
Evaluate lim+ 𝑥 ln 𝑥.
𝑥→0
Indeterminate Differences
If lim 𝑓 𝑥 − 𝑔 𝑥
𝑥→𝑎
produces the indeterminate
form ∞ − ∞, then try rewriting the difference as
a quotient by:
Getting a
common
denominator
Rationalizing
Factoring
Exercise 7
Evaluate
1
lim+
𝑥→1 ln 𝑥
−
1
.
𝑥−1
Indeterminate Powers
If lim 𝑓 𝑥
𝑥→𝑎
𝑔 𝑥
produces one of the
indeterminate forms 00 , ∞0 , or 1∞ , then
try logarithmic differentiation.
Exercise 8
Evaluate lim 1 +
𝑥→∞
1 𝑥
.
𝑥
Exercise 9
Evaluate lim+ sin 𝑥 𝑥 .
𝑥→0
Summary
Determinate Forms
∞+∞→∞
−∞ − ∞ → −∞
0∞ → 0
0−∞ → ∞
Indeterminate Forms
0 ∞
,
0 ∞
0∙∞
∞−∞
00 , ∞0 , 1∞
Use L’H
Rewrite as
a quotient
Take the log
8-7: “L’Hôpital’s” Rule
Objectives:
1. To apply “L’Hôpital’s”
Rule to evaluate limits
Assignment
• P.574-576: 11-51 eoo,
59, 60, 63, 64, 65-69
odd, 75, 91-94, 97