Page 62 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 53

Then the graph, shown in the adjoining Fig. 3-12, f (x)
consists of the lines y ¼ 1, x > 3; y ¼ 1, x < 3 and
the point ð3; 0Þ.
(b)As x ! 3 from the right, f ðxÞ! 1, i.e., lim f ðxÞ¼ 1,
1
x!3þ
as seems clear from the graph. To prove this we must
x
show that given any > 0, we can find > 0suchthat (3, 0)
j f ðxÞ 1j < whenever 0 < x 1 < . 1
Now since x > 1, f ðxÞ¼ 1 and so the proof con-
sists in the triviality that j1 1j < whenever
0 < x 1 < .
Fig. 3-12
(c) As x ! 3 from the left, f ðxÞ! 1, i.e.,
lim f ðxÞ¼ 1. A proof can be formulated as in (b).
x!3
(d)Since lim f ðxÞ 6¼ lim f ðxÞ, lim f ðxÞ does not exist.
x!3
x!3þ x!3
3.13. Prove that lim x sin 1=x ¼ 0.
x!0
We must show that given any > 0, we can find > 0suchthat jx sin 1=x 0j < when
0 < jx 0j < .
If 0 < jxj < ,then jx sin 1=xj¼jxjj sin 1=xj @ jxj < since j sin 1=xj @ 1for all x 6¼ 0.
Making the choice ¼ ,we see that jx sin 1=xj < when 0 < jxj < ,completing the proof.
2
3.14. Evaluate lim 1=x .
x!0þ 1 þ e
As x ! 0þ we suspect that 1=x increases indefinitely, e 1=x increases indefinitely, e 1=x approaches 0,
1 þ e 1=x approaches 1; thus the required limit is 2.
To prove this conjecture we must show that, given > 0, we can find > 0suchthat

2

2 < when 0 < x <
1 þ e 1=x

2 2 2 2e 1=x 2
Now 2 ¼ ¼

1 þ e 1=x 1 þ e 1=x e 1=x þ 1
Since the function on the right is smaller than 1 for all x > 0, any > 0will work when e 1. If
2 e 1=x þ 1 1 2 1 2 1
0 < < 1, then < when > , e 1=x > 1, > ln 1 ;or0 < x < ¼ .
e 1=x þ 1 2 x lnð2= 1Þ
1
3.15. Explain exactly what is meant by the statement lim ¼1 and prove the validity of this
4
statement. x!1 ðx 1Þ
The statement means that for each positive number M,wecan find a positive number (depending on
M in general) such that
1
4 > 4 when 0 < jx 1j <
ðx 1Þ
1 1 1
4
To prove this note that > M when 0 < ðx 1Þ < ffiffiffiffiffi.
4 M or 0 < jx 1j < p 4
ðx 1Þ M
p ffiffiffiffiffi
4
Choosing ¼ 1= M,the required results follows.
sin
3.16. Present a geometric proof that lim ¼ 1.
!0
Construct a circle with center at O and radius OA ¼ OD ¼ 1, as in Fig. 3-13 below. Choose point B on
OA extended and point C on OD so that lines BD and AC are perpendicular to OD.
It is geometrically evident that