2.8. USING DERIVATIVES TO EVALUATE LIMITS
153
a
g
m = g ′ (a)
f
m = f ′ (a)
a
m = g ′ (a)
m = f ′ (a)
Figure 2.20: Two functions f and g that satisfy L’Hopital’s Rule.
matters most, but rather the rate at which each approaches zero that determines the value
of the limit. This is a good way to remember what L’Hopital’s Rule says: if f (a) = g(a) = 0,
the the limit of
f (x)
g(x) as x → a is given by the ratio of the slopes of f and g at x = a.
Activity 2.23.
In this activity, we reason graphically from the following figure to evaluate limits of
ratios of functions about which some information is known.
1
2
3
4
-2
-1
1
2
f
g
1
2
3
4
-2
-1
1
2
p
q
1
2
3
4
-2
-1
1
2
s
r
Figure 2.21: Three graphs referenced in the questions of Activity 2.23.
(a) Use the left-hand graph to determine the values of f (2), f ′ (2), g(2), and g ′ (2).
Then, evaluate
lim
x→2
f (x)
g(x)
.
(b) Use the middle graph to find p(2), p ′ (2), q(2), and q ′ (2). Then, determine the
153
a
g
m = g ′ (a)
f
m = f ′ (a)
a
m = g ′ (a)
m = f ′ (a)
Figure 2.20: Two functions f and g that satisfy L’Hopital’s Rule.
matters most, but rather the rate at which each approaches zero that determines the value
of the limit. This is a good way to remember what L’Hopital’s Rule says: if f (a) = g(a) = 0,
the the limit of
f (x)
g(x) as x → a is given by the ratio of the slopes of f and g at x = a.
Activity 2.23.
In this activity, we reason graphically from the following figure to evaluate limits of
ratios of functions about which some information is known.
1
2
3
4
-2
-1
1
2
f
g
1
2
3
4
-2
-1
1
2
p
q
1
2
3
4
-2
-1
1
2
s
r
Figure 2.21: Three graphs referenced in the questions of Activity 2.23.
(a) Use the left-hand graph to determine the values of f (2), f ′ (2), g(2), and g ′ (2).
Then, evaluate
lim
x→2
f (x)
g(x)
.
(b) Use the middle graph to find p(2), p ′ (2), q(2), and q ′ (2). Then, determine the