278
CHAPTER
4

Applications
of
Derivatives
SOLUTION
Because
k—0and
1
—
¢™*
—
0,
we can
apply
I'Hospital's
Rule:
,
A
1
—e*
.
et
lim
C
=
lim
r
»
==
rlim
—
k—0
k—0
k
0

=p]=p
Thus
the
upper
bound
for
the
concentration
is
r,
the
same
as
the
injection
rate.
U
BT

Find
lim
————
=

—cosx
SOLUTION
If
we
blindly
attempted
to
use
I'Hospital's
Rule,
we
would
get
sin
x
_
Cos
X
lim
———=
lim
—
=
—o¢
t—=x

—cosXx
x—=
sinx
This
is
wrong!
Although
the
numerator
sin
x
—
0
as
x
—
7,
notice
that
the
denomi
nator
(1
—
cos
x)
does
not
approach
0,
so
I'Hospital's
Rule
can't
be
applied
here.
The
required
limit
is,
in
fact,
easy
to
find
because
the
function
is
continuous
at
7
and
the
denominator
is
nonzero
there:
_
sin
x
sin
7
0
lim
—
—
=0
B
v—=a

—
cos
x
l
—cosm
1
—(1)
Example
5
shows
what
can
go
wrong
if
you
use
I'Hospital's
Rule
without
thinking.
Other
limits
can
be
found
using
I'Hospital's
Rule
but
are
more
easily
found
by
other
methods.
(See
Examples
2.4.3,
24.5,
and
2.2.5
and
the
discussion
at
the
beginning
of
this
section.)
So
when
evaluating
any
limit,
you
should
consider
other
methods
before
using
I'Hospital's
Rule.
B
Which
Functions
Grow
Fastest?
L'Hospital's
Rule
enables
us
to
compare
the
rates
of
growth
of
functions.
Suppose
we
have
two
functions
f(x)
and
g(x)
that
both
become
large
as
x
becomes
large,
that
is,
lim
f(x)
=
=
and
lim
g(x)
=
=
—x
We
say
that
f(x)
approaches
infinity
more
quickly
than
g(x)
if
For
example,
we
used
I'Hospital's
Rule
in
Example
2
to
show
that
.
e
lim
—
=
=
r—em
X
and
so
the
exponential
function
y
=
¢*
grows
more
quickly
than
y
=
x°,
In
fact
y
=
¢*
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