Page 64 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 55

(b)We have
j f ðxÞgðxÞ ABj¼j f ðxÞ½gðxÞ BŠþ B½ f ðxÞ AŠj ð4Þ
@ j f ðxÞjjgðxÞ BjþjBjj f ðxÞ Aj
@ j f ðxÞjjgðxÞ BjþðjBjþ 1Þj f ðxÞ Aj
Since lim f ðxÞ¼ A,wecan find 1 such j f ðxÞ Aj < 1for 0 < jx x 0 j < 1 , i.e.,
x!x 0
A 1 < f ðxÞ < A þ 1, so that f ðxÞ is bounded, i.e., j f ðxÞj < P where P is a positive constant.
Since lim gðxÞ¼ B,given > 0wecan find 2 > 0suchthat jgðxÞ Bj < =2P for
x!x 0
0 < jx x 0 j < 2 .
Since lim f ðxÞ¼ A,given > 0wecan find 3 > 0suchthat j f ðxÞ Aj < for
x!x 0
2ðjBjþ 1Þ
0 < jx x 0 j < 2 .
Using these in (4), we have

¼
j f ðxÞgðxÞ ABj < P þðjBjþ 1Þ
2P 2ðjBjþ 1Þ
for 0 < jx x 0 j < where is the smaller of 1 ; 2 ; 3 and the proof is complete.
(c) We must show that for any > 0wecan find > 0suchthat

1
1 jgðxÞ Bj
< when 0 < jx x 0 j <
¼ ð5Þ
B
gðxÞ jBjjgðxÞj
By hypothesis, given > 0we can find 1 > 0suchthat
2
jgðxÞ Bj < B when 0 < jx x 0 j < 1
1
2
By Problem 3.18, since lim gðxÞ¼ B 6¼ 0, we can find 2 > 0suchthat
x!x 0
1 when
2
jgðxÞj > jBj 0 < jx x 0 j < 2
Then if is the smaller of 1 and 2 ,wecan write
1 2
1 B
1 jgðxÞ Bj 2
< ¼ whenever 0 < jx x 0 j <
¼ 1
B
2
gðxÞ jBjjgðxÞj
jBj jBj
and the required result is proved.
(d)From parts (b) and (c),
1 1 1 A
lim f ðxÞ ¼ lim f ðxÞ ¼ lim f ðxÞ lim ¼ A ¼
x!x 0 x!x 0 B B
x!x 0 gðxÞ
gðxÞ x!x 0 gðxÞ
This can also be proved directly (see Problem 3.69).
The above results can also be proved in the cases x ! x 0 þ, x ! x 0 , x !1, x ! 1.
Note:Inthe proof of (a)we have used the results j f ðxÞ Aj < =2 and jgðxÞ Bj < =2, so that the final
result would come out to be j f ðxÞþ gðxÞ ðA þ BÞj < .Ofcourse the proof would be just as valid if we
had used 2 (or any other positive multiple of )inplace of .A similar remark holds for the proofs of ðbÞ,
(c), and (d).
3.20. Evaluate each of the following, using theorems on limits.
2
2
limðx 6x þ 4Þ¼ lim x þ limð 6xÞþ lim 4
ðaÞ
x!2 x!2 x!2 x!2
¼ðlim xÞðlim xÞþðlim 6Þðlim xÞþ lim 4
x!2 x!2 x!2 x!2 x!2
¼ð2Þð2Þþð 6Þð2Þþ 4 ¼ 4
In practice the intermediate steps are omitted.