The Real And Complex Number Systems

f x fx .
So, as x M, wehave
e x e M e M fM e x
e x fx
e x e x e M e M fM .
If we let e x e M e M fM gx, ande x e M e M fM hx, then we have
g x e x fx h x
and
gM e M fM hM.
Hence, for x M,
e x e M e M fM gx
e x fx
hx e x e M e M fM
It implies that, for x M,
e x e M e M fM fx e x e M e M fM
which implies that
lim fx 0since is arbitrary.
x
Note: Intheprocessofproof,weusetheresultonMean Value Theorem. Letf, g, and
h be continuous on a, b and differentiable on a, b. Suppose fa ga ha and
f x g x h x on a, b. Show that fx gx hx on a, b.
Proof: By Mean Value theorem, wehave
gx fx ga fa gx fx
g c f c, wherec a, x.
0 by hypothesis.
So, fx gx on a, b. Similarly for gx hx on a, b. Hence, fx gx hx
on a, b.
(vii) Let fx a 1 sin x ...a n sin nx, wherea i are real for i 1, 2, . . n. Suppose that
|fx| |x| for all real x. Prove that |a 1 ..na n| 1.
Proof: Let x 0, and by Mean Value Theorem, we have
|fx f0| |fx| |a 1 sin x ...a n sin nx|
|f cx|, wherec 0, x
|a 1 cosc ...na n cosncx|
|x| by hypothesis.
So,
|a 1 cosc ...na n cosnc| 1
Note that as x 0 ,wehavec 0 ; hence, |a 1 ..na n| 1.
Note: Here are another type:
(a) |sin 2 x sin 2 y| |x y| for all x, y.
(b) |tan x tan y| |x y| for all x, y , . 2 2
(viii) Let f : R R be differentiable with f x c for all x, wherec 0. Show that