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Mathematics LibreTexts

4.4.E: Problems on Limits and Operations in E

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Exercise 4.4.E.1

Show by examples that all expressions (1) are indeterminate.

Exercise 4.4.E.2

Give explicit definitions for the following "unsigned infinity" limit statements:
 (a) limxpf(x)=; (b) limxp+f(x)=;(c)limxf(x)=.

Exercise 4.4.E.3

Prove at least some of Theorems 110 and formulas (i)(iv) in Note 1.

Exercise 4.4.E.4

In the following cases, find limf(x) in two ways: (i) use definitions only; (ii) use suitable theorems and justify each step accordingly.
 (a) limx1x(=0). (b) limxx(x1)13x2 (c) limx2+x22x+1x23x+2 (d) limx2x22x+1x23x+2 (e) limx2x22x+1x23x+2(=)
[Hint: Before using theorems, reduce by a suitable power of x.]

Exercise 4.4.E.5

Let
f(x)=nk=0akxk and g(x)=mk=0bkxk(an0,bm0).
Find limxf(x)g(x) if (i)n>m;( ii )n<m; and (iii) n=m(n,mN).

Exercise 4.4.E.6

Verify commutativity and associativity of addition and multiplication in E, treating Theorems 116 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.]

Exercise 4.4.E.7

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 z0. Show by examples that it may fail in other cases; e.g., if x=y=+, z=1.


4.4.E: Problems on Limits and Operations in E is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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