MA121, Homework #4
due Monday, Jan. 21 in class
1. Compute each of the following limits:
x3 + x2 − 5x + 3
L2 = lim 3
.
x→1 x − x2 − x + 1
log x
L1 = lim
,
x→∞
x
2. Compute each of the following limits:
M1 = lim
x→0
3x − 1
,
x
M2 = lim (ex + x)1/x .
x→0
3. Let f be the function defined by
½
f (x) =
0
1
if x 6= 0
if x = 0
¾
.
Show that f is integrable on [0, 1].
4. Suppose f, g are both integrable on [a, b] and f (x) ≤ g(x) for all x ∈ [a, b]. Show that
Z
Z
b
b
f (x) dx ≤
a
g(x) dx.
a
• You are going to work on these problems during your Friday tutorials.
• When writing up solutions, write legibly and coherently. Use words, not just symbols.
• Write both your name and your tutor’s name on the first page of your homework.
• Your tutor’s name is Derek, if you are a TP student; otherwise, it is Pete.
• Your solutions may use any of the axioms/results stated in class (but nothing else).
• NO LATE HOMEWORK WILL BE ACCEPTED.
Hints and comments
1. Those are ∞/∞ and 0/0 limits, so one may use L’Hôpital’s rule.
2. The first limit is a 0/0 limit, so one may use L’Hôpital’s rule to get
M1 = lim
x→0
3x − 1
(3x )0
= lim
= lim (3x )0 .
x→0
x→0
x
1
To compute the derivative on the right hand side, try to get rid of the exponent:
f (x) = 3x
=⇒
log f (x) = log 3x = x log 3
1
· f 0 (x) = log 3.
f (x)
=⇒
For the second limit, you can use a similar approach to get
M2 = lim (ex + x)1/x
x→0
=⇒
log M2 = lim
x→0
log(ex + x)
.
x
Note that the rightmost limit is a 0/0 one and then use L’Hôpital’s rule.
3. To show that f is integrable on [0, 1], you have to show that
sup {S − (f, P )} = inf {S + (f, P )}.
P
P
Start with a partition P = {x0 , x1 , . . . , xn } and try to first show that
S − (f, P ) = 0,
S + (f, P ) = x1 .
4. By definition, the integral is equal to the supremum of the lower Darboux sums. Thus,
you need to show that
sup {S − (f, P )} ≤ sup {S − (g, P )}.
P
P
Forget about the suprema for the moment and try to show that
S − (f, P ) ≤ S − (g, P )
for all partitions P . Once you have shown this, you may then take the sup of both sides
to finish the proof. Taking the inf/sup of both sides is known to preserve the inequality.