The Real And Complex Number Systems

Proof: Trivially, f x 1/x 2 1
cos1/x, ifx 0. Let x n
2n 1 2
consider
fx n f0
sin1/x n
x n 0 x n
2n 1 as n .
2
Hence, we know that f 0 does not exist.
(b) g x sin1/x 1/x cos1/x, ifx 0; g 0 does not exist.
, and thus
Proof: Trivially, g 1
x sin1/x 1/x cos1/x, ifx 0. Let x n ,and
2n 1 2
y n 1 , we know that 2n
gx n g0
sin 1
x n 0 xn
1 for all n
and
gy n g0
sin 1
y n 0 yn
0 for all n.
Hence, we know that g 0 does not exist.
(c) h x 2x sin1/x cos1/x, ifx 0; h 0 0; lim x0 h x does not exist.
Proof: Trivially, h x 2x sin1/x cos1/x, ifx 0. Consider
hx h0
|x sin1/x| |x| 0asx 0,
x 0
so we know that h 1
0 0. In addition, let x n ,andy
2n 1 n 1 ,wehave
2n
2
h x n 2
2n 1 and h y n 1 for all n.
2
Hence, we know that lim x0 h x does not exist.
5.6 Derive Leibnitz’s formula for the nth derivative of the product h of two functions
f and g :
n
h n kn f k g nk x, where kn n!
k!n k! .
k0
Proof: We prove it by mathematical Induction. As n 1, it is clear since
h f g g f. Suppose that n k holds, i.e., h k k
j0
jk f j g kj x. Consider
n k 1, we have