140 Chapter3
3. The infinitesimal J(z) = z^3 + 3z^4 is of higher order than g(z) = z^2 + z
(as z ---t 0) since limz_,.O f(z)/g(z) = 0. Equivalently, g(z) is of lower
order than J(z).Theorem 3.4 If two infinitesimals f(z) and g(z) as z ---t a differ by an
infinitesimal h(z) of higher order than g(z) as z ---t a, then J(z) ,..., g(z)
as z ---t a.
Proof If J(z) = g(z) + h(z), then
lim f(z) = lim [i + h(z)) = 1
z_,.a g( z) z_,.a g( z)Theorem 3.5 The limit as z ---+ a of a function F(z) that contains the
infinitesimal J(z) as a factor (or divisor) does not change if f(z) is replaced
by an equivalent infinitesimal g(z) (as z ---ta), assuming that J(z) -1- 0 in
N~(a) n Dt for some 8 > 0.
Proof Suppose that F(z) = '/;j and limz_,.a F(z) = L. Replacing f(z)
by g(z), we get
lim g(z) = lim [ f(z) · g(z)] = L(l) = L
z_,.a h( z) z-+a h( Z) J ( Z)
The two preceding theorems are very useful in computing the limits of
expressions involving infinitesimals. According to these theorems, if any
factor or divisor of the given expression contains a sum of infinitesimals
of different orders, those of higher order can be discarded without altering
the limit to be sought.
Example
. 3z^2 + z^3 + z^5 3-^2
hm = lim ....:.:__ = 3
z-+O z^2 + 2z^4 + z^6 z-+O z^2
Bachmann-Landau notations. Bachmann introduced, and Landau
popularized, the following symbols, called the little-oh and the big-oh sym-
bols, which are helpful in abbreviating certain analytical expressions: If
limz-+a f(z)/g(z) = 0, we write f(z) = o {g(z)} as z ---t a. This is read
"f(z) is little-oh of g(z)," or "f(z) is an infinitesimal of higher order than
g(z)" as z approaches a (this value is sometimes to be understood from
the context).
If IJ(z)I :::; Klg(z)I, K a constant, when z is restricted to a certain
subset of Dt n D 9 , or when z is made to approach a certain value a (an
accumulation point of Dt n D 9 ), we write f(z) = 0 {g(z)}, which is read
"f(z) is big-oh of g(z)" or "f(z) is of the order of g(z)" for z in the above-
mentioned subset of Dt n D 9 , or as z ---t a.