1. Homework 7 (1) Use definition to prove that
(a) lim
x→π2−tan x = ∞.
(b) lim
x→−π2+tan x = −∞.
(c) lim
x→0+
√x = 0.
(2) Evaluate (a) lim
x→∞x p3
x3+ x2+ 1 −p3
x3− x2+ 1 .
(b) lim
x→0+
v u u t1
x+ s
1 x+
r1 x−
v u u t1
x+ s
1 x−
r1 x
. (3) Let
f (x) =
(√x − 4 if x > 4 8 − 2x if x < 4.
Determine whether lim
x→4f (x) exists.
(4) Find a, b such that lim
x→∞
x2+ 1
x + 1 − ax − b
= 0.
(5) For what value of the constant c is the function f continuous on R?
f (x) =
(cx2+ 2x if x < 2, x3− cx if x ≥ 2.
(6) For each x ∈ R, we define
∞
X
n=0
xn n!.
You can use the ratio test to show that the infinite series is convergent for all x ∈ R. Hence R → R, x 7→
∞
X
n=0
xn n!
defines a real valued function on R. This function is denoted by ex. Let f (x) =
(e−1/x2 if x 6= 0,
0 x = 0.
Find lim
x→0f (x) and lim
x→∞f (x).
(7) Let (xn) be a sequence of real numbers such that lim
n→∞xn= ∞. Suppose f (x) is a function defined on [a, ∞) such that lim
x→∞f (x) = L. Show that lim
n→∞f (xn) = L.
(8) Let f (x) ∈ C(R). If f (x1+ x2) = f (x1) + f (x2) for all x1, x2∈ R, show that f(x) = f(1)x, for x ∈ R. Use the following steps:
(a) Show that f (n) = nf (1) for any n ∈ N by induction.
(b) Show that f (0) = 0 and f (−x) = −f (x) for any x ∈ R.
(c) Use (a) and (b) to conclude that f (m) = f (1)m for any m ∈ Z.
(d) Use (c) to show that f (r) = f (1)r for any r ∈ Q.
(e) Use (d) and the fact that any real number is the limit of a sequence of rational numbers to conclude that f (x) = f (1)x is true for any x ∈ R.
(9) Let C[a, b] be the set of all functions f : [a, b] → R continuous on [a, b]. Let f, g, h ∈ C[a, b], and k, a, b ∈ R.
(i) We define function f + g : [a, b] → R by (f + g)(x) = f (x) + g(x) for x ∈ [a, b].
(ii) We define function k · f : [a, b] → R by (k · f )(x) = kf (x) for x ∈ [a, b].
1
2
(iii) We define the zero function O : [a, b] → R by O(x) = 0 for all x ∈ [a, b].
(iv) We define the (−f ) : [a, b] → R by (−f )(x) = −f (x) for x ∈ [a, b].
(v) We define f g : [a, b] → R by (f g)(x) = f (x)g(x) for x ∈ [a, b].
(vi) If g 6= O, we define f /g : [a, b] → R by (f /g)(x) = f (x)/g(x) for x ∈ [a, b].
Prove that f + g, k · f, −f, O, f g ∈ C[a, b] and f /g ∈ C[a, b] when g 6= O and the followings:
(a) f + g = g + f.
(b) (f + g) + h = f + (g + h).
(c) O + f = f + O = f.
(d) f + (−f ) = (−f ) + f = 0.
(e) (ab) · f = a(b · f ).
(f) (a + b) · f = a · f + b · f.
(g) a · (f + g) = a · f + a · g.
(h) 1 · f = f.
