l’Hôpital’s Rule & Indeterminate Forms
As you work through the problems listed below, you should reference Chapter 3.6 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know how to use l’Hôpital’s Rule to help compute limits involving indeterminate forms
∞
0
of and
0
∞
• Be able to compute limits involving indeterminate forms ∞ − ∞, 0 · ∞, 00 , ∞0 , and
1∞ by manipulating the limits into a form where l’Hôpital’s Rule is applicable.
PRACTICE PROBLEMS:
For problems 1-27, calculate the indicated limit. If a limit does not exist, write
+∞, −∞, or DNE (whichever is most appropriate). Make sure that l’Hôpital’s
Rule applies before using it. And, whenever you apply l’Hôpital’s Rule, indicate
that you are doing so.
x2 + 4x − 21
x→3 x2 − 7x + 12
−10
1. lim
tan 3x
x→0 ln (1 + x)
2. lim
3
sin x − x
x→0
x2
0
3. lim
sin (6x)
x→0 sin (3x)
4. lim
2
5. lim−
x→1
x−1
x2 − 2x + 1
−∞
e−x
x→∞ x−2
0
6. lim
1
ln (cos 3x)
x→0
5x2
9
−
10
7. lim
tan−1 x −
x→1
x−1
8. lim
π
4
1
2
√
8 x−1
√
9. lim+
x→0 1 − 5 x
−
3 ln 2
ln 5
5 sin x
10. lim+ √
x→0
x
0
x3 + 4x − 5
x→−∞ 5x2 − 5x − 89
−∞
11. lim
sin−1 (2x)
12. lim
x→0 tan−1 (3x)
2
3
ln x2
x→1 x2 − 9
13. lim
0
ex − e−x
x→∞ ex + e−x
14. lim
1
ln x − π2
15. lim
tan x
x→ π2 +
0
2
x−1
arccos x
16. lim−
x→1
0
√
e
17. lim
x→+∞
x
x
+∞
18. lim xe−6x
x→+∞
0
√
19. lim
x→+∞
4 + 3x2
2 + 2x
√
3
2
20. lim+ x csc 3x
x→0
1
3
21. lim [ln (x + 2) − ln (3x + 5)]
x→+∞
1
ln
3
22. lim 3x 7−x
x→∞
0
23. lim
x→∞
−
√
x2 − x − x
1
2
24. lim+ tan x sec x
x→0
0
25. lim+ x1/x
x→0
0
3
5x
2
26. lim 1 +
x→∞
x
e10
−x
1
27. lim 1 −
x→∞
x
e
28. Which of the following are indeterminate forms?
0
0
∞−∞
00
∞
0
0
∞
∞+∞
∞0
0∞
∞
∞
0·∞
1∞
∞∞
0 ∞
,
, ∞ − ∞, 0 · ∞, 00 , ∞0 , 1∞
0 ∞
29. Calculate each of the following limits:
(a) lim+ (1 + 3x )1/x
x→0
+∞
(b) lim− (1 + 3x )1/x
x→0
0
(c) lim (1 + 3x )1/x
x→+∞
3
(d) lim (1 + 3x )1/x
x→−∞
1
xn
= 0 for any positive integer n.
x→∞ ex
30. Show that lim
4
∞·∞
∞1
xn
∞
, so, we may apply L’Hopital’s Rule:
is of the indeterminate form
x
x→∞ e
∞
lim
xn
nxn−1
=
lim
x→∞ ex
x→∞
ex
lim
∞
This new limit is also of the indeterminate form , so, we may again apply L’Hopital’s
∞
Rule:
nxn−1
n(n − 1)xn−2
lim
=
lim
x→∞
x→∞
ex
ex
In fact, we repeat the process until we end up with the following limit:
n(n − 1)(n − 2) . . . (2)(1)
x→∞
ex
lim
xn
=0
x→∞ ex
which equals 0. Thus, lim
31. Find the value(s) of of the constant k which make f (x) =

sin x − 1
π


if x 6=

 x− π
2
2



 k
continuous at x =
π
.
2
k=0
32. Find all values of k and m such that lim
x→1
k = 0 and m = 5
33. Multiple Choice: What is lim+
x→1
x
?
ln(x)
(a) 0
(b) 1
(c) e
(d) e−1
(e) +∞
E
5
k + m ln x
=5
x−1
if x =
π
2
ex − 1
?
x→0 tan(x)
34. Multiple Choice: What is lim
(a) −1
(b)
0
(c)
1
(d)
2
(e)
The limit does not exist.
C
35. Multiple Choice: If lim f (x) = lim g(x) = +∞ and f 0 (x) = 1 and g 0 (x) = ex ,
x→+∞
x→+∞
f (x)
?
what is lim
x→+∞ g(x)
(a) −1
(b)
0
(c)
1
(d)
e
(e)
The limit does not exist.
B
36. A Regular Polygon is a polygon that is equiangular (all angles are equal in measure)
and equilateral (all sides have the same length). The diagram below shows several
regular polygons inscribed within a circle of radius r.
6
(a) Let An be the area of a regular n-sided polygon inscribed within a circle of radius
2π
r. Divide the polygon into n congruent triangles each with a central angle of
n
radians, as shownin the
diagram
above
for
several
different
values
of
n.
Show
1 2
2π
that An = r sin
n.
2
n
We begin by examining one of the n triangles, pictured below.
The base of the triangle has a length of r. And, the height of the triangle is r sin θ,
2π
where θ is the central angle,
. Thus, the area of one triangle is:
n
1
2π
1 2
2π
A = (r) r sin
= r sin
2
n
2
n
But, the polygon is composed of n such triangles. So, the area of a regular n-sided
polygon inscribed in the circle of radius r is:
1 2
2π
An = r sin
n
2
n
(b) What can you conclude about the area of the n-sided polygon as the number of
sides of the polygon, n, approaches infinity? In other words, compute lim An .
n→∞
lim An = πr
2
n→∞
7