8.2 L’Hôpital’s Rule Quick Review What you’ll learn about Indeterminate Form 0/0...
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Transcript of 8.2 L’Hôpital’s Rule Quick Review What you’ll learn about Indeterminate Form 0/0...
8.2
L’Hôpital’s Rule
Quick Review
0
1
1
2
0
Use graphs or tables to estimate the value of the limit.
11. lim 1
12. lim 1
13. lim
1
4 14. lim
1tan
5. lim2 tan
x
x
x
x
t
x
x
x
x
t
t
x x
xx
x
1
2
2
0
Quick Review
1Substitute to express as a function of .
16. sin
17. 1
x
x y hh
y xx
yx
h
hy
sin
hhy /11
What you’ll learn about Indeterminate Form 0/0 Indeterminate Forms ∞/∞, ∞·0, ∞-∞ Indeterminate Form 1∞, 00, ∞0
Essential QuestionHow can limits be used to describe thebehavior of functions?How do we use l’Hôpital’s Rule in finding limits?
Indeterminate Form 0/0
If functions f (x) and g (x) are both zero at x = a, then xg
xfax
lim
cannot be found by substituting x = a. The substitution produces 0/0, a meaningless expression known as an indeterminate form.
L’Hôpital’s Rule (First Form)
Suppose that f (a) = g (a) = 0, that f ’ (a) and g’ (a) exist, and that
.limThen .0 ag
af
xg
xfag
ax
Example Indeterminate Form 0/01. Use L’Hôpital’s Rule to find the limit.
x
xx
24lim
0
24 xxf 240 f 0 xxg 00 g
xf 2
1
4 x 1 xg
x
xx
24lim
0
1
421
lim2
1
0
x
x
xx
42
1lim
0 4
1
2
1
L’Hôpital’s Rule (Stronger Form)Suppose that f (a) = g (a) = 0, that f and g are differentiable on an open interval I containing a, and that g’ (x) ≠ 0 on I if x ≠ a. Then
.limlim xg
xf
xg
xfaxax
Example Using L’Hôpital’s Rule with One-Sided Limits
2. Evaluate using L’Hôpital’s Rule:x
xx 2
sinlim
0
00sin0 f 0020 g
x
xx 2
sinlim
0
lim
0
x
xcos2 2
1
Example Working with Indeterminate Form ∞/∞
3. Identify the indeterminate form and evaluate the limit using L’Hôpital’s Rule.
x
xx cot1
csclim
Since the numerator and denominator are discontinuous at x = , we can look at one-sided limits there.
cscf cot1g
x
xx cot1
csclim
lim
x
xx cotcscx2csc x
xx csc
cotlim
lim
x
xcosxsin
xsin1
xx
coslim
1
Since the right-hand limit is also equal – 1, then the two-sided limit is equal – 1.
Example Working with Indeterminate Form ∞·0
.1
sin2lim Find 4.
xx
x
2f
1
sing 00
xh
1Let
xx
x
1sin2lim
lim
0h
2h
h sin
h
hh
sin2lim
0
lim
0
h
h cos21
2
Indeterminate Forms 1∞, 00,∞0
Here a can be finite or infinite.
Lxfax
ln lim Lxf
axaxeexf
lnlimlim
Example Working with Indeterminate Form 1∞
.1
1lim Find 5.x
x x
x
xxf
11Let
xxxf
11lnln
x1
1ln
x
1
This is the indeterminate form 0/0.
Apply L’Hôpital’s Rule:
xfx
lnlim
xx
11ln
lim
x
1
lim
x
x1
1
1
2
1
x
2
1
x
x
x 11
1lim
1
Therefore,
x
x x
11lim xf
xlnlim
xf
xelnlim
1e e
Example Working with Indeterminate Form 00
.lim Find 6.0
x
xx
xxxf Let
xxxf lnln xln
x
1Apply L’Hôpital’s Rule:
xfx
lnlim0
xx
lnlim
0
x
1
lim
0
x
x
1
2
1
x
xx
0
lim 0
Therefore,x
xx
0lim xf
xlnlim
0 xf
xeln
0lim
0e 1
Example Working with Indeterminate Form ∞0
.lim Find 7.1
x
xx
xxxf
1
Let
xx
xf ln1
ln xln xApply L’Hôpital’s Rule:
xfx
lnlim x
xx
lnlim
lim
x
x
1
10
Therefore, x
xx
1
lim
xfx
lnlim
xf
xelnlim
0e 1
Pg. 450, 8.2 #1-51 odd