Homework 6
Math 15300 (section 51), Spring 2015
This homework is due in class on Wednesday, May 13th. You may cite results from class as appropriate.
Unless otherwise stated, you must provide a complete explanation for your solutions, not simply an
answer. You are encouraged to work together on these problems, but you must write up your solutions
independently.
You are encouraged to think about the problems marked with a (*), but they are not to be handed in.
0. (*) Read sections 11.5-7 in the text.
1. Calculate the limits:
ex − 1
x→0 ln(x + 1)
ex − 1
(d) (Ex 11.5.10 ) lim
x→0 x(1 + x)
√
1 − x2
(f) (Ex 11.5.23) lim √
x→1−
1 − x3
cos x − cos 3x
(h) (Ex 11.5.27) lim
x→0
sin(x2 )
1
(j) (Ex 11.5.35) lim
n→∞ n [ln(n + 1) − ln n]
ln x
x→1 x − 1
x−a
(c) (Ex 11.5.6) lim n
x→a x − an
cos x − 1 + x2 /2
(e) (Ex 11.5.18) lim
x→0
x4
ln(sin x)
(g) (Ex 11.5.25) lim
x→π/2 (π − 2x)2
(b) (Ex 11.5.3) lim
(a) (Ex 11.5.2) lim
(i) (Ex 11.5.33) lim
n→∞
π/2 − arctan n
1/n
2. Calculate the limits:
x3 − 1
x→∞ 2 − x
(c) (Ex 11.6.8) lim (x ln | sin x|)
ln xk
x→∞ x
ln x
(d) (Ex 11.6.12) lim
+
x→0 cot x
(f) (Ex 11.6.18) lim xsin x
(a) (Ex 11.6.4) lim
(b) (Ex 11.6.6) lim
x→0
(e) (Ex 11.6.16) lim | sin x|x
x→0
1
1
(g) (Ex 11.6.21) lim
−
x→0 ln(1 + x)
x
x→0+
(h) (Ex 11.6.32) lim (ex + 3x)1/x
x→0
3. Determine whether the (improper) integrals converge. If they do, then evaluate the integral.
Z ∞
Z ∞
dx
(a) (Ex 11.7.3)
4
+
x2
0
Z 8
dx
(c) (Ex 11.7.7)
2/3
0 x
Z ∞
dx
(e) (Ex 11.7.14)
x ln x
Ze ∞
x
(g) (Ex 11.7.26)
dx
(1
+
x2 )2
1
Z 4
dx
(i) (Ex 11.7.30)
2
1 x − 5x + 6
(b) (Ex 11.7.4)
e−px dx, p > 0
0
Z 1
dx
1 − x2
0
Z ∞
dx
(f) (Ex 11.7.18)
2
x −1
2
Z ∞
dx
(h) (Ex 11.7.28)
x
−x
−∞ e + e
(d) (Ex 11.7.9)
√
Z ∞
(j) (Ex 11.7.31)
0
1
e−x sin xdx
4. (Ex 11.5.44) Show that if a > 0 then lim n a1/n − 1 = ln a.
n→∞
sin 2x + ax + bx3
=0
x→0
x3
5. (Ex 11.5.46) Find the values of a and b for which lim
6. (Ex 11.5.48) Let f be a twice differentiable function, and fix a constant x.
f (x + h) − f (x − h)
= f 0 (x).
h→0
2h
f (x + h) − 2f (x) + f (x − h)
= f 00 (x).
(b) Show that lim
h→0
h2
(a) Show that lim
7. (Ex 11.6.57) What is wrong with the following computation:
lim
x→0+
x2
2x
2
= lim
= lim
= −∞?
+
+
sin x x→0 cos x x→0 − sin x
Z ∞
Z b
sin xdx diverges, but lim
8. (Ex 11.7.58) Show that
b→∞ −b
−∞
sin xdx = 0.
9. Use the comparison test to determine whether the integral converges:
Z ∞
(a) (Ex 11.7.51)
1
Z ∞
(c) (Ex 11.7.55)
1
Z ∞
x
dx
1 + x5
ln x
dx
x2
√
(b) (Ex 11.7.53)
(1 + x5 )−1/6
0
Z ∞
(d) (Ex 11.7.56)
e
1
10. (*) (Ex 11.7.60) The function f (x) = √
2π
integral converges for all real x.
Z x
2 /2
e−t
dx
√
x + 1 ln x
dt is important in statistics1 . Show that this
−∞
11. The gamma function. For any s > 0, define
Z ∞
Γ(s) =
xs−1 e−x dx.
0
This is know as the gamma function. It shows up frequently both in mathematics, and in science
(especially in physics and statistics).
(a) (*) Show that the improper integral above converges for any value of s. Z(Hint: Comparison
∞
test. Show that, for some a, e−x < x−(s+1) for all x > a. Then compare
xs−1 e−x dx with
a
Z ∞
x
−2
dx.)
a
(b) Use integration by parts to show that Γ(s + 1) = sΓ(s) for all s > 0. (Hint: u = xs ,
dv = e−x dx.)
(c) Show that Γ(1) = 1.
(d) Prove by induction that Γ(n) = (n − 1)! for all positive integers n.
This means that the gamma function can be thought of as a generalization of the factorial function:
For any positive real2 , we can define s! = Γ(s + 1).
1
2
Specifically, its the cumulative distribution function for the normal distribution.
In fact, there’s a way to do this for negative s, or even for any complex s, except for s = −1, −2, −3 . . ..
2