3.3. L’HÔPITAL’S RULE; STRANGE POLYNOMIALS 107
Problem 3.33 Using the product rule, prove the statement: If p.x/ is a polynomial and x D a
is a root of p.x/ with multiplicity k > 1, then x D a is also a root of f
0
.x/ (from Knowledge
Box 3.6).
Problem 3.34 Using material from this section, prove that
y D cos.x/
is not a polynomial function.
Problem 3.35 Suppose two polynomials share the same roots and, for those roots, the same
multiplicities. If the polynomials are not equal, how do they differ?
3.3 L’HÔPITAL’S RULE; STRANGE POLYNOMIALS
During our study of curve sketching, we developed a rule of thumb about computing the limit
at infinity of the ratio of two polynomials. If p.x/ D ax
n
C is a polynomial of degree n and
q.x/ D bx
m
C is a polynomial of degree m the rule said:
lim
x!1
p.x/
q.x/
D
8
<
:
˙1 n > m
a=b n D m
0 n < m
is rule is a consequence of L’Hôpital’s rule, a useful tool for computing limits that cannot be
resolved other ways.
Knowledge Box 3.7
L’Hôpital’s rule at x D c
If lim
x!c
f .x/ D 0 and lim
x!c
g.x/ D 0
or
if lim
x!c
f .x/ D ˙1 and lim
x!c
g.x/ D ˙1, then
lim
x!c
f .x/
g.x/
D lim
x!c
f
0
.x/
g
0
.x/
108 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
e hypotheses that the limits of the numerator and denominator are both going to zero or
both going to some sort of infinity are critical. L’Hôpital’s rule will give the wrong answer if
the hypotheses are not satisfied.
Let’s do an example.
Example 3.36 Compute
lim
x!2
x
2
4
x 2
with L’Hôpital’s rule.
Solution:
We could resolve this limit by factoring and canceling, but, since the top and bottom of
the fraction are both 0 at x D 2, L’Hôpital’s rule applies.
lim
x!2
x
2
4
x 2
D lim
x!2
2x
1
D
4
1
D 4
˙
Knowledge Box 3.8
L’Hôpital’s rule as x ! 1
If lim
x!1
f .x/ D 0 and lim
x!1
g.x/ D 0
or
if lim
x!1
f .x/ D ˙1 and lim
x!1
g.x/ D ˙1, then
lim
x!1
f .x/
g.x/
D lim
x!1
f
0
.x/
g
0
.x/
is version of L’Hôpital’s rule is the one that gives us the rule-of-thumb for the limit at
infinity of ratios of polynomials.
3.3. L’HÔPITAL’S RULE; STRANGE POLYNOMIALS 109
Let’s do an example.
Example 3.37 Compute
lim
x
!1
x
2
C 2x C3
3x
2
C 1
:
Solution:
Both the numerator and denominator are going to infinity, so:
lim
x!1
x
2
C 2x C3
3x
2
C 1
D lim
x!1
2x C2
6x
L’Hôpital’s rule still applies
D lim
x!1
2
6
D
1
3
And so the limit at infinity is
1
3
– exactly what the rule of thumb gives us.
˙
e reason the rule of thumb works for polynomials of equal degree is that L’Hôpital’s rule
applies until we have taken so many derivatives that the numerator and denominator are con-
stants, at which point the accumulated values from bringing the power out front cancel, leaving
the ratio of the highest degree coefficients as the limit.
At this point, we add a name to our mathematical vocabulary with the following definition.
Definition 3.2 A rational function is a function that is the ratio of polynomial functions.
We have been working with and graphing rational functions for some time now, so they are not
new. We just have a name for them now. While L’Hôpital’s rule was introduced to back-fill
the rule of thumb for limits at infinity of rational functions (to use their new name), it in fact
applies to all continuous, differentiable functions.
Let’s do a couple of examples.
110 3. THE ARITHMETIC, GEOMETRY, AND CALCULUS OF POLYNOMIALS
Example 3.38 Compute
lim
x!0
sin.x/
x
:
Solution:
e numerator and denominator both go to zero, so L’Hôpital’s rule applies.
lim
x!0
sin
.x/
x
D
lim
x!0
cos.x/
1
D
1
˙
Example 3.39 Compute
lim
x!0
cos.x/ 1
x
:
Solution:
e numerator and denominator are both going to zero, so L’Hôpital’s rule applies.
lim
x!0
cos
.x/
1
x
D lim
x!0
sin.x/
1
D 0
˙
As with every other technique we have learned, we may use algebraic rearrangement to permit
us to extend the reach of L’Hôpital’s rule.
Example 3.40 Compute
lim
x!0
x ln.x/:
Solution:
is problem does not fit the form for L’Hôpital’s rule, but we can coerce it into that
3.3. L’HÔPITAL’S RULE; STRANGE POLYNOMIALS 111
form:
lim
x!0
x ln.x/ D lim
x!0
ln.x/
1
x
Now in correct form
1
1
for L’Hôpital
D lim
x!0
1
x
1
x
2
Use L’Hôpital
D lim
x!0
x
2
x
D lim
x!0
x
1
Simplify
D 0 Done
So x moves toward zero faster than ln.x/ moves toward negative infinity as x ! 0.
˙
It is possible for L’Hôpital to take you to a limit that clearly does not exist. As before, the rules-
of-thumb developed for curve sketching can also supply the answer, but in mathematics a solid
method is preferred.
Example 3.41 Find lim
x!1
x
3
C 1
x
2
.
Solution:
e rule-of-thumb tells us this diverges to 1, but let’s work through this with L’Hôpital’s rule
and see where it goes.
lim
x!1
x
3
C 1
x
2
D lim
x!1
3x
2
2x
D lim
x!1
3
2
x ! 1
L’Hôpital’s rules also tells us this diverges to infinity.
˙
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