NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA
1505 Mathematics
I
Tutorial 2 (Solution Notes)
1. Use L’Hopital’s rule to find the following limits.
(a)
lim
x
→
π/
2
1
−
sin
x
1 + cos 2
x
(b) lim
x
→
0
ln(cos
ax
)
ln(cos
bx
)
,
a, b >
0
(c) lim
x
→∞
x
tan
1
x
(d)
lim
x
→
0+
x
a
ln
x
,
a >
0
(e) lim
x
→
1
x
1
1
−
x
(f)
lim
x
→
0
+
x
sin
x
(g) lim
x
→
0
sin
x
x
1
x
2
Preliminaries
The Limit Laws
If
L
,
M
,
c
and
k
are real numbers and
lim
x
→
c
f
(
x
) =
L
and
lim
x
→
c
g
(
x
) =
M,
then
1.
Sum Rule
:
lim
x
→
c
[
f
(
x
) +
g
(
x
) ] =
L
+
M
The limit of the sum of two functions is the sum of their limits.
6
2.
Difference Rule
:
lim
x
→
c
[
f
(
x
)
−
g
(
x
) ] =
L
−
M
The limit of the difference of two functions is the difference of their limits.
7
3.
Product Rule
:
lim
x
→
c
[
f
(
x
)
·
g
(
x
) ] =
L
·
M
The limit of a product of two functions is the product of their limits.
8
4.
Constant Multiple Rule
:
lim
x
→
c
[
k
·
f
(
x
) ] =
k
·
L
The limit of a constant times a function is the constant times the limit of the
function.
6
If
L
+
M
is of the form
∞
+ (
−∞
), then this rule does not hold any more, and you need to consider what is
f
(
x
) +
g
(
x
).
7
If
L
−
M
is of the form
∞−∞
, then this rule does not hold any more, and you need to consider what is
f
(
x
)
−
g
(
x
).
8
If
L
·
M
is of the form 0
· ∞
, then you need to use L’Hopital’s rule instead.
16
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5.
Quotient Rule
:
lim
x
→
c
[
f
(
x
)
g
(
x
)
] =
L
M
The limit of a quotient of two functions is the quotient of their limits, provided
the limit of the denominator is not zero.
9
6.
Power Rule
: If
r
and
s
are integers with no common factor and
s
= 0 then
lim
x
→
c
[
f
(
x
) ]
r/s
=
L
r/s
provided that
L
r/s
is a real number. (If
s
is even, we assume that
L >
0.)
The limit of a rational power of a function is that power of the limit of the
function, provided the latter is a real number.
L’Hopital’s Rule
The rule is named after the 17thcentury French mathematician Guillaume de L’Hopital, who
published the rule in his book l’Analyse des Infiniment Petits pour l’Intelligence des Lignes
Courbes (literal translation: Analysis of the Infinitely Small to Understand Curved Lines)
(1696), the first book about differential calculus.
L’Hospital’s Rule
Suppose
and
are differentiable and
near
(except
possibly at ). Suppose that
and
or that
and
(In other words, we have an indeterminate form of type
or
.) Then
if the limit on the right side exists (or is
or
).
lim
x
l
a
f x
t
x
lim
x
l
a
f
x
t
x
0
0
lim
x
l
a
t
x
lim
x
l
a
f x
lim
x
l
a
t
x
0
lim
x
l
a
f x
0
a
a
t
x
0
t
f
Remarks:
•
L’Hospital’s Rule is also valid for onesided limits and for limits at infinity
or negative infinity; that is, “
x
→
a
”can be replaced by any of the symbols
x
→
a
+
,
x
→
a
−
,
x
→ ∞
, or
x
→ −∞
.
•
L’Hospital’s Rule says that the limit of a quotient of functions is equal to the
limit of the quotient of their derivatives, provided that the given conditions are
satisfied. It is especially important to verify the conditions regarding the limits
of and before using L’Hospital’s Rule.
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 Spring '09
 MA1505
 Calculus, Derivative, Sin, Cos, lim g

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