4.4.E: Problems on Limits and Operations in E∗
( \newcommand{\kernel}{\mathrm{null}\,}\)
Show by examples that all expressions (1∗) are indeterminate.
Give explicit definitions for the following "unsigned infinity" limit statements:
(a) limx→pf(x)=∞; (b) limx→p+f(x)=∞;(c)limx→∞f(x)=∞.
Prove at least some of Theorems 1−10 and formulas (i)−(iv) in Note 1.
In the following cases, find limf(x) in two ways: (i) use definitions only; (ii) use suitable theorems and justify each step accordingly.
(a) limx→∞1x(=0). (b) limx→∞x(x−1)1−3x2 (c) limx→2+x2−2x+1x2−3x+2 (d) limx→2−x2−2x+1x2−3x+2 (e) limx→2x2−2x+1x2−3x+2(=∞)
[Hint: Before using theorems, reduce by a suitable power of x.]
Let
f(x)=n∑k=0akxk and g(x)=m∑k=0bkxk(an≠0,bm≠0).
Find limx→∞f(x)g(x) if (i)n>m;( ii )n<m; and (iii) n=m(n,m∈N).
Verify commutativity and associativity of addition and multiplication in E∗, treating Theorems 1−16 and formulas (2∗) as definitions. Show by examples that associativity and commutativity (for three terms or more) would fail if, instead of (2∗), the formula (±∞)+(∓∞)=0 were adopted.
[Hint: For sums, first suppose that one of the terms in a sum is +∞; then the sum is + ∞. For products, single out the case where one of the factors is 0; then consider the infinite cases.]
Continuing Problem 6, verify the distributive law (x+y)z=xz+yz in E∗, assuming that x and y have the same sign (if infinite), or that z≥0. Show by examples that it may fail in other cases; e.g., if x=−y=+∞, z=−1.