 8.2.1: Show that the sequence xn /(1 + xn does not converge uniformly on [...
 8.2.2: Prove that the sequence in Example 8.2.1(c) is an example of a sequ...
 8.2.3: Construct a sequence of functions on [0, 1] each of which is discon...
 8.2.4: Suppose Un) is a sequence of continuous functions on an interval I ...
 8.2.5: Let I : R + R be uniformly continuous on R and let In(x) := I(x + ...
 8.2.6: LetIn(x) := 1/(1 + xt forx e [0, 1].FindthepointwiselimitI of the s...
 8.2.7: Suppose the sequence Un) converges uniformly to I on the set A, and...
 8.2.8: Let In(x) := nx/(1 + nx2) for x e A := [0,00), Show that each In is...
 8.2.9: Let In (X) := Xn/n for x E [0,1]. Show that the sequ~nce (In) of di...
 8.2.10: Let gn(x) := enx /n for x ::: 0, n E N. Examine the relation betwe...
 8.2.11: Let I := [a, b] and let Un) be a sequence of functions on I + IRth...
 8.2.12: Show that lim fl2 enx2 dx = O.
 8.2.13: If a > 0, show that limL7r(sinnx)/(nx) dx = O.What happens if a = O?
 8.2.14: Let In(x) := nx/(1 + nx) for x E [0, 1]. Show that Un) converges no...
 8.2.15: Letgn(x):= nx(1 x)n for x E [0,1], n EN. Discuss the convergence o...
 8.2.16: Let {rl, r2, . . . , rn . . .} be an enumeration of the rational nu...
 8.2.17: Let In (x) := 1for x E (0, l/n) and In (x) :=Oelsewherein[O, 1].Sho...
 8.2.18: Let In(x) := xn for x E [0, 1], n EN. Show that Un) is a decreasing...
 8.2.19: Let In(x) := x/n forx E [0,00), n EN. Show that Un) is a decreasing...
 8.2.20: Give an example of a decreasing sequence Un) of continuous function...
Solutions for Chapter 8.2: Interchange of Limits
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 8.2: Interchange of Limits
Get Full SolutionsSince 20 problems in chapter 8.2: Interchange of Limits have been answered, more than 6321 students have viewed full stepbystep solutions from this chapter. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.2: Interchange of Limits includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3.

Additive identity for the complex numbers
0 + 0i is the complex number zero

Arcsecant function
See Inverse secant function.

Arrow
The notation PQ denoting the directed line segment with initial point P and terminal point Q.

Circular functions
Trigonometric functions when applied to real numbers are circular functions

Direction of an arrow
The angle the arrow makes with the positive xaxis

equation of a parabola
(x  h)2 = 4p(y  k) or (y  k)2 = 4p(x  h)

Focal width of a parabola
The length of the chord through the focus and perpendicular to the axis.

Frequency table (in statistics)
A table showing frequencies.

Geometric sequence
A sequence {an}in which an = an1.r for every positive integer n ? 2. The nonzero number r is called the common ratio.

Interval
Connected subset of the real number line with at least two points, p. 4.

kth term of a sequence
The kth expression in the sequence

Line of symmetry
A line over which a graph is the mirror image of itself

Linear function
A function that can be written in the form ƒ(x) = mx + b, where and b are real numbers

Maximum rvalue
The value of r at the point on the graph of a polar equation that has the maximum distance from the pole

NDER ƒ(a)
See Numerical derivative of ƒ at x = a.

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Power function
A function of the form ƒ(x) = k . x a, where k and a are nonzero constants. k is the constant of variation and a is the power.

Terms of a sequence
The range elements of a sequence.

Unit circle
A circle with radius 1 centered at the origin.

Vertical line test
A test for determining whether a graph is a function.