2.8. USING DERIVATIVES TO EVALUATE LIMITS
151
h(x) that can be written in the form h(x) =
f (x)
g(x) where f and g are both differentiable at
x = a and for which f (a) = g(a) = 0. We are interested in finding a way to evaluate the
indeterminate limit given by lim
x→a
h(x). In Figure 2.19, we see a visual representation of the
situation involving such functions f and g. In particular, we see that both f and g have
an x-intercept at the point where x = a. In addition, since each function is differentiable,
each is locally linear, and we can find their respective tangent line approximations L f and
L g at x = a, which are also shown in the figure. Since we are interested in the limit of
f (x)
g(x) as x → a, the individual behaviors of f (x) and g(x) as x → a are key to understand.
Here, we take advantage of the fact that each function and its tangent line approximation
become indistinguishable as x → a.
First, let’s reall that L f (x) = f ′ (a)(x − a) + f (a) and L g (x) = g ′ (a)(x − a) + g(a). The
critical observation we make is that when taking the limit, because x is getting arbitrarily
close to a, we can replace f with L f and replace g with L g , and thus we observe that
lim
x→a
f (x)
g(x)
= lim
x→a
L f (x)
L g (x)
= lim
x→a
f ′ (a)(x − a) + f (a)
g ′ (a)(x − a) + g(a)
.
Next, we remember a key fundamental assumption: that both f (a) = 0 and g(a) = 0, as
this is precisely what makes the original limit indeterminate. Substituting these values for
f (a) and g(a) in the limit above, we now have
lim
x→a
f (x)
g(x)
= lim
x→a
f ′ (a)(x − a)
g ′ (a)(x − a)
= lim
x→a
f ′ (a)
g ′ (a)
,
where the latter equality holds since x is approaching (but not equal to) a, so
x−a
x−a = 1.
Finally, we note that
f ′ (a)
g ′ (a) is constant with respect to x, and thus
lim
x→a
f (x)
g(x)
=
f ′ (a)
g ′ (a)
.
We have, of course, implicitly made the assumption that g ′ (a) 0, which is essential to
the overall limit having the value
f ′ (a)
g ′ (a) . We summarize our work above with the statement
of L’Hopital’s Rule, which is the formal name of the result we have shown.
L’Hopital’s Rule: Let f and g be differentiable at x = a, and suppose that
f (a) = g(a) = 0 and that g ′ (a) 0. Then lim x→a
f (x)
g(x) =
f ′ (a)
g ′ (a) .
In practice, we typically work with a slightly more general version of L’Hopital’s Rule,
which states that (under the identical assumptions as the boxed rule above and the extra
151
h(x) that can be written in the form h(x) =
f (x)
g(x) where f and g are both differentiable at
x = a and for which f (a) = g(a) = 0. We are interested in finding a way to evaluate the
indeterminate limit given by lim
x→a
h(x). In Figure 2.19, we see a visual representation of the
situation involving such functions f and g. In particular, we see that both f and g have
an x-intercept at the point where x = a. In addition, since each function is differentiable,
each is locally linear, and we can find their respective tangent line approximations L f and
L g at x = a, which are also shown in the figure. Since we are interested in the limit of
f (x)
g(x) as x → a, the individual behaviors of f (x) and g(x) as x → a are key to understand.
Here, we take advantage of the fact that each function and its tangent line approximation
become indistinguishable as x → a.
First, let’s reall that L f (x) = f ′ (a)(x − a) + f (a) and L g (x) = g ′ (a)(x − a) + g(a). The
critical observation we make is that when taking the limit, because x is getting arbitrarily
close to a, we can replace f with L f and replace g with L g , and thus we observe that
lim
x→a
f (x)
g(x)
= lim
x→a
L f (x)
L g (x)
= lim
x→a
f ′ (a)(x − a) + f (a)
g ′ (a)(x − a) + g(a)
.
Next, we remember a key fundamental assumption: that both f (a) = 0 and g(a) = 0, as
this is precisely what makes the original limit indeterminate. Substituting these values for
f (a) and g(a) in the limit above, we now have
lim
x→a
f (x)
g(x)
= lim
x→a
f ′ (a)(x − a)
g ′ (a)(x − a)
= lim
x→a
f ′ (a)
g ′ (a)
,
where the latter equality holds since x is approaching (but not equal to) a, so
x−a
x−a = 1.
Finally, we note that
f ′ (a)
g ′ (a) is constant with respect to x, and thus
lim
x→a
f (x)
g(x)
=
f ′ (a)
g ′ (a)
.
We have, of course, implicitly made the assumption that g ′ (a) 0, which is essential to
the overall limit having the value
f ′ (a)
g ′ (a) . We summarize our work above with the statement
of L’Hopital’s Rule, which is the formal name of the result we have shown.
L’Hopital’s Rule: Let f and g be differentiable at x = a, and suppose that
f (a) = g(a) = 0 and that g ′ (a) 0. Then lim x→a
f (x)
g(x) =
f ′ (a)
g ′ (a) .
In practice, we typically work with a slightly more general version of L’Hopital’s Rule,
which states that (under the identical assumptions as the boxed rule above and the extra