4.4: Infinite Limits. Operations in E*
( \newcommand{\kernel}{\mathrm{null}\,}\)
As we have noted, Theorem 1 of §3 does not apply to infinite limits, even if the function values f(x),g(x),h(x) remain finite (i.e., inE1). Only in certain cases (stated below) can we prove some analogues.
There are quite a few such separate cases. Thus, for brevity, we shall adopt a kind of mathematical shorthand. The letter q will not necessarily denote a constant; it will stand for
"a function f:A→E1,A⊆(S,ρ), such that f(x)→q∈E1 as x→p."
Similarly, "0" and "±∞" will stand for analogous expressions, with q replaced by 0 and ±∞, respectively.
For example, the "shorthand formula" (+∞)+(+∞)=+∞ means
"The sum of two real functions, with limit+∞ at p (p∈S), is itself a function with limit+∞ at p."
The point p is fixed, possibly ±∞(if A⊆E∗). With this notation, we have the following theorems.
1. (±∞)+(±∞)=±∞.
2. (±∞)+q=q+(±∞)=±∞.
3. (±∞)⋅(±∞)=+∞.
4. (±∞)⋅(∓∞)=−∞.
5. |±∞|=+∞.
6. (±∞)⋅q=q⋅(±∞)=±∞ if q>0.
7. (±∞)⋅q=q⋅(±∞)=∓∞ if q<0.
8. −(±∞)=∓∞.
9. (±∞)q=(±∞)⋅1q if q≠0.
10. q(±∞)=0.
11. (+∞)+∞=+∞.
12. (+∞)−∞=0.
13. (+∞)q=+∞ if q>0.
14. (+∞)q=0 if q<0.
15. If q>1, then q+∞=+∞ and q−∞=0.
16. If 0<q<1, then q+∞=0 and q−∞=+∞.
- Proof
-
We prove Theorems 1 and 2, leaving the rest as problems. (Theorems 11-16 are best postponed until the theory of logarithms is developed.)
1. Let f(x) and g(x)→+∞ as x→p. We have to show that
f(x)+g(x)→+∞,
i.e., that
(∀b∈E1)(∃δ>0)(∀x∈A∩G¬p(δ))f(x)+g(x)>b
(we may assume b>0). Thus fix b>0. As f(x) and g(x)→+∞, there are δ′,δ′′>0 such that
(∀x∈A∩G¬p(δ′))f(x)>b and (∀x∈A∩G¬p(δ′′))g(x)>b.
Let δ=min(δ′,δ′′). Then
(∀x∈A∩G¬p(δ))f(x)+g(x)>b+b>b,
as required; similarly for the case of −∞.
2. Let f(x)→+∞ and g(x)→q∈E1. Then there is δ′>0 such that for x in A∩G¬p(δ′),|q−g(x)|<1, so that g(x)>q−1.
Also, given any b∈E1, there is δ′′ such that
(∀x∈A∩G−p(δ′′))f(x)>b−q+1.
Let δ=min(δ′,δ′′). Then
(∀x∈A∩G¬p(δ))f(x)+g(x)>(b−q+1)+(q−1)=b,
as required; similarly for the case of f(x)→−∞.
Caution: No theorems of this kind exist for the following cases (which therefore are called indeterminate expressions):
(+∞)+(−∞),(±∞)⋅0,±∞±∞,00,(±∞)0,00,1±∞.
In these cases, it does not suffice to know only the limits of f and g. It is necessary to investigate the functions themselves to give a definite answer, since in each case the answer may be different, depending on the properties of f and g. The expressions (1*) remain indeterminate even if we consider the simplest kind of functions, namely sequences, as we show next.
(a) Let
um=2m and vm=−m.
(This corresponds to f(x)=2x and g(x)=−x.) Then, as is readily seen,
um→+∞,vm→−∞, and um+vm=2m−m=m→+∞.
If, however, we take xm=2m and ym=−2m, then
xm+ym=2m−2m=0;
thus xm+ym is constant, with limit 0 (for the limit of a constant function equals its value; see §1, Example (a)).
Next, let
um=2m and zm=−2m+(−1)m.
Then again
um→+∞ and zm→−∞, but um+zm=(−1)m;
um+zm "oscillates" from −1 to 1 as m→+∞, so it has no limit at all.
These examples show that (+∞)+(−∞) is indeed an indeterminate expression since the answer depends on the nature of the functions involved. No general answer is possible.
(b) We now show that 1+∞ is indeterminate.
Take first a constant {xm},xm=1, and let ym=m. Then
xm→1,ym→+∞, and xymm=1m=1=xm→1.
If, however, xm=1+1m and ym=m, then again ym→+∞ and xm→1 (by Theorem 10 above and Theorem 1 of Chapter 3, §15), but
xymm=(1+1m)m
does not tend to 1; it tends to e>2, as shown in Chapter 3, §15. Thus again the result depends on {xm} and {ym}.
In a similar manner, one shows that the other cases (1*) are indeterminate.
Note 1. It is often useful to introduce additional "shorthand" conventions. Thus the symbol ∞ (unsigned infinity) might denote a functionf such that
|f(x)|→+∞ as x→p;
we then also write f(x)→∞. The symbol 0+ (respectively, 0−) denotes a function f such that
f(x)→0 as x→p
and, moreover
f(x)>0 (f(x)<0, respectively) on some G¬p(δ).
We then have the following additional formulas:
(i) (±∞)0+=±∞,(±∞)0−=∓∞.
(ii) If q>0, then q0+=+∞ and q0−=−∞.
(iii) ∞0=∞.
(iv) q∞=0.
The proof is left to the reader.
Note 2. All these formulas and theorems hold for relative limits, too.
So far, we have defined no arithmetic operations in E∗. To fill this gap (at least partially), we shall henceforth treat Theorems 1-16 above not only as certain limit statements (in "shorthand") but also as definitions of certain operations in E∗. For example, the formula (+∞)+(+∞)=+∞ shall be treated as the definition of the actual sum of +∞ and +∞ in E∗, with +∞ regarded this time as an element of E∗ (not as a function). This convention defines the arithmetic operations for certain cases only; the indeterminate expressions (1*) remain undefined, unless we decide to assign them some meaning.
In higher analysis, it indeed proves convenient to assign a meaning to at least some of them. We shall adopt these (admittedly arbitrary) conventions:
{(±∞)+(∓∞)=(±∞)−(±∞)=+∞;00=1;0⋅(±∞)=(±∞)⋅0=0 (even if 0 stands for the zero-vector ).
Caution: These formulas must not be treated as limit theorems (in "short-hand"). Sums and products of the form (2*) will be called "unorthodox."