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Mathematics LibreTexts

4.4: Infinite Limits. Operations in E*

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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:AE1,A(S,ρ), such that f(x)qE1 as xp."

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 (pS), is itself a function with limit+ at p."

The point p is fixed, possibly ±(if AE). With this notation, we have the following theorems.

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 q0.

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 xp. We have to show that

f(x)+g(x)+,

i.e., that

(bE1)(δ>0)(xAG¬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

(xAG¬p(δ))f(x)>b and (xAG¬p(δ))g(x)>b.

Let δ=min(δ,δ). Then

(xAG¬p(δ))f(x)+g(x)>b+b>b,

as required; similarly for the case of .

2. Let f(x)+ and g(x)qE1. Then there is δ>0 such that for x in AG¬p(δ),|qg(x)|<1, so that g(x)>q1.

Also, given any bE1, there is δ such that

(xAGp(δ))f(x)>bq+1.

Let δ=min(δ,δ). Then

(xAG¬p(δ))f(x)+g(x)>(bq+1)+(q1)=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.

Examples

(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=2mm=m+.

If, however, we take xm=2m and ym=2m, then

xm+ym=2m2m=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

xm1,ym+, and xymm=1m=1=xm1.

If, however, xm=1+1m and ym=m, then again ym+ and xm1 (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 xp;

we then also write f(x). The symbol 0+ (respectively, 0) denotes a function f such that

f(x)0 as xp

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."


This page titled 4.4: Infinite Limits. Operations in E* is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Elias Zakon (The Trilla Group (support by Saylor Foundation)) via source content that was edited to the style and standards of the LibreTexts platform.

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