Page 54 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 45

6. Inverse hyperbolic functions. If x ¼ sinh y then y ¼ sinh 1 x is the inverse hyperbolic sine of x.
The following list gives the principal values of the inverse hyperbolic functions in terms of
natural logarithms and the domains for which they are real.

p ffiffiffiffiffiffiffiffiffiffiffiffiffi !
2
p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 x þ 1
1
2
sinh x þ 1 Þ; all x csch x ¼ ln ; x 6¼ 0
ðaÞ x ¼ lnðx þ ðdÞ þ
x jxj
!
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 1 x 2
1
2
cosh x 1 Þ; x A 1 ðeÞ sech x ¼ ln ; 0 < x @ 1
x
ðbÞ x ¼ lnðx þ

1 1 þ x 1 x þ 1
tanh 1 x ¼ ln ; jxj < 1 coth 1 x ¼ ln ; jxj > 1
2 1 x 2 x 1
ðcÞ ð f Þ
LIMITS OF FUNCTIONS
Let f ðxÞ be defined and single-valued for all values of x near x ¼ x 0 with the possible exception of
x ¼ x 0 itslef (i.e., in a deleted neighborhood of x 0 ). We say that the number l is the limit of f ðxÞ as x
approaches x 0 and write lim f ðxÞ¼ l if for any positive number (however small) we can find some
x!x 0
positive number (usually depending on ) such that j f ðxÞ lj < whenever 0 < jx x 0 j < .In such
case we also say that f ðxÞ approaches l as x approaches x 0 and write f ðxÞ! l as x ! x 0 .
In words, this means that we can make f ðxÞ arbitrarily close to l by choosing x sufficiently close to
x 0 .
2
x if x 6¼ 2
EXAMPLE. . Then as x gets closer to 2 (i.e., x approaches 2), f ðxÞ gets closer to 4. We
Let f ðxÞ¼
0if x ¼ 2
thus suspect that lim f ðxÞ¼ 4. To prove this we must see whether the above definition of limit (with l ¼ 4) is
x!2
satisfied. For this proof see Problem 3.10.
Note that lim f ðxÞ 6¼ f ð2Þ, i.e., the limit of f ðxÞ as x ! 2is not the same as the value of f ðxÞ at x ¼ 2 since
x!2
f ð2Þ¼ 0bydefinition. The limit would in fact be 4 even if f ðxÞ were not defined at x ¼ 2.
When the limit of a function exists it is unique, i.e., it is the only one (see Problem 3.17).
RIGHT- AND LEFT-HAND LIMITS
In the definition of limit no restriction was made as to how x should approach x 0 .Itis sometimes
found convenient to restrict this approach. Considering x and x 0 as points on the real axis where x 0 is
fixed and x is moving, then x can approach x 0 from the right or from the left. We indicate these
respective approaches by writing x ! x 0 þ and x ! x 0 .
If lim f ðxÞ¼ l 1 and lim f ðxÞ¼ l 2 ,we call l 1 and l 2 , respectively, the right- and left-hand limits of
x!x 0 þ x!x 0
f at x 0 and denote them by f ðx 0 þÞ or f ðx 0 þ 0Þ and f ðx 0 Þ or f ðx 0 0Þ. The ; definitions of limit of
f ðxÞ as x ! x 0 þ or x ! x 0 are the same as those for x ! x 0 except for the fact that values of x are
restricted to x > x 0 or x < x 0 , respectively.
We have lim f ðxÞ¼ l if and only if lim f ðxÞ¼ lim f ðxÞ¼ l.
x!x 0 x!x 0 þ x!x 0

THEOREMS ON LIMITS
If lim f ðxÞ¼ A and lim gðxÞ¼ B, then
x!x 0 x!x 0
1: lim ð f ðxÞþ gðxÞÞ ¼ lim f ðxÞþ lim gðxÞ¼ A þ B
x!x 0 x!x 0 x!x 0