ConcepTest • Section 4.7 • Question 1

True or false?

xgxfxgxfxg af

xQxQxgxfxQagaf

ag

af

ag

afagaf

axax

ax

axax

).(')('lim)()(limThen .)(limand 0)( Suppose (c)

).('lim)(limThen ).(/)()(Let .0)()( Suppose (b)

.)('

)('

)(

)( rule,

sHopital'l' toAccording .0)()( Suppose (a)

ConcepTest • Section 4.7 • Answer 1

ANSWER

COMMENT:Follow-up Question. What does l’Hopital’s rule say?Answer. If f and g are differentiable, f (a) = g(a) = 0, and g’(a) ≠ 0, then

(a) False. l’Hopital’s rule concerns the equality of limits, not the equality of functions.

(b) False. In applying l’Hopital’s rule, the numerator and the denominator are differentiated separately.

(c) False. To apply l’Hopital’s rule to a product, the expression must first be rewritten as a ratio.

.)('

)('

)(

)(lim

ag

af

xg

xfax

Consider 3 limits:

Which is the correct ranking?(a)p < q < r(b)p < r < q(c) q < p < r(d)q < r < p(e)r < p < q(f) r < q < p

ConcepTest • Section 4.7 • Question 2

44

122lim

23

23

1

xxx

xxxp

x 26

7lim

3

3

1

xx

xxq

x 45

sinlim

21

xx

xr

x

ConcepTest • Section 4.7 • Answer 2

119

9

26

7lim

3

3

1

xx

xxq

x

ANSWER(a) Since p is a limit of the form 0/0 it can be

evaluated by l’Hopital’s rule.

The limit for q is the limit of a function that is continuous

at x = 1, so l’Hopital’s rule does not apply. To compute the

limit, evaluate the function.

2

1

10

5

183

243lim

44

122lim

2

2

123

23

1

xx

xx

xxx

xxxp

xx

Since r is a limit of the form 0/0 it can be evaluated by l’Hopital’s rule.

We have

COMMENT:Students may attempt to evaluate q with l’Hopital’s

rule. Remindthem that l’Hopital’s rule usually gives the wrong

answer when it does not apply.ConcepTest • Section 4.7 • Answer 2

.3352

coslim

45

sinlim

121

x

x

xx

xr

xx

ANSWER (cont’d)

31

2

1 rqp

ConcepTest • Section 4.7 • Question 3

For which of the following can you use l’Hopital’s rule to evaluate the limit?

2

0x

2

x

0x

0x

lim (d)

lim (c)

sinlim (b)

coslim (a)

xe

xex

xx

x

x

x

ConcepTest • Section 4.7 • Answer 3

ANSWER

COMMENT:Follow-up Question. Compute the limits.Answer. For (a) the limit does not exist; for (b) the limit is 1; for (c) and (d) the limit is zero.

.lim asit rewritingafter (c) (b),2

x xe

x

Consider 3 limits:

Which is the correct ranking?(a)p < q < r(b)p < r < q(c) q < p < r(d)q < r < p(e)r < p < q(f) r < q < p

ConcepTest • Section 4.7 • Question 4

20

cos1lim

x

xp

x

x

xq

x

lnlim

.)(lim 22 xx

xeexr

(c) Since p is a limit of the form 0/0 it can be evaluated by l’Hopital’s rule. The first application of the rule leads to another limit of the form 0/0, so l’Hopital’s rule must be applied a second time.

Since q is a limit of the form ∞/∞ it can be evaluated by

l’Hopital’s rule.

ConcepTest • Section 4.7 • Answer 4

02

lim)2/1(

/1lim

lnlim

2/1

xx

x

x

xq

xxx

ANSWER

.2

1

2

coslim

2

sinlim

cos1lim

0020

x

x

x

x

xp

xxx

Since r is a limit of the form ∞*0 it can be rewritten in the form

∞/∞, and then l’Hopital’s rule can be applied.

We have

COMMENT:The limits for q and r could be the starting point for a

discussion of dominance relations among logarithmic, power, and

exponential functions.ConcepTest • Section 4.7 • Answer 4

.11012

1lim

2

21limlim)(lim

22

2

2

222

xxx

x

xx

x

x

xx

x ee

e

e

exeexr

ANSWER (cont’d)

12

10 rpq

ConcepTest • Section 4.7 • Question 5

Arrange in order by dominance as x ∞, from least to most dominant.

xexx

exx

x

x

x

ln (e) (d)

(c)

)(ln (b) (a)

2

2

2

100

ConcepTest • Section 4.7 • Answer 5

ANSWER

(b), (d), (a), (e), (c). As x ∞, in increasing order of dominance, we have

x(ln x)2 < x2 < x100 < ex lnx < e2x / x.

COMMENT:Do the students know the answer without using l’Hopital’s Rule?