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