Math 166Z Homework # 6
Due Thursday March 10th
1. Evaluate the following limits or explain why they do not exist:
(a)
lim (sec x − tan x)
x→(π/2)−
(b) lim+ (1 + sin 4t)cot t
t→0
(c) lim+ xx
x→0
(d) lim e−x ln x
x→∞
2. Evaluate the following integrals or explain why they are divergent:
Z 1
xe2x dx
(a)
−∞
Z ∞
2
(b)
xe−x dx
−∞
Z ∞
ln x
(c)
dx
x3
1
Z ∞
1
(d)
dx
x2
1
Z 3
1
(e)
dt
0 t−1
Z 1
(f)
ln x dx
0
Z
2
(g)
−3
1
dx
x4
3. The Laplace Transform of a function f (t) is defined as follows:
L {f (t)} =
Z
∞
e−st f (t) dt
0
Notice that it changes functions of t into functions of a new variable s. It is useful for
solving differential equations (if you take Math 267 you will see it again) but for now it
is just useful to get some practice with improper integrals. Find:
(a) L {t}
(b) L {et }
Bonus question: Find:
(a) L {sin t}
(b) L {g(t)}


0≤t<5
2,
where g(t) = 0,
5 ≤ t < 10

 4t
e , 10 ≤ t
(c) L {t2 + 2t + 1}
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