- 4.1.1: Determine a condition on lx - 11 that will assure that: (a) lx 2 - ...
- 4.1.2: Determine a condition on lx - 41 that will assure that: (a) I.Ji- 2...
- 4.1.3: Let c be a cluster point of As; :Rand let f: A~ :R. Prove that lim ...
- 4.1.4: Let f: :R ~:Rand let c e :R. Show that lim f(x) = L if and only if ...
- 4.1.5: Let I := (0, a) where a > 0, and let g(x) := x2 for x e I. For any ...
- 4.1.6: Let I be an interval in :R, let f : I ~ :R, and let c e I. Suppose ...
- 4.1.7: Show that lim .JX = .jC for any c > x-c
- 4.1.8: Show that lim .JX = .jC for any
- 4.1.9: Use either the e-~ definition of limit or the Sequential Criterion ...
- 4.1.10: Use the definition of limit to show that (a) lim(x2 + 4x) = 12, x-2 1
- 4.1.11: Show that the following limits do not exist. (a) lim ~ (x > 0), x-o...
- 4.1.12: Suppose the function f : R ~ :R has limit L at 0, and let a > H g :...
- 4.1.13: Let c e :Rand let f: :R ~ :R be such that lim(f<x>) 2 = L. x-c (a) ...
- 4.1.14: Let f : :R ~ :R be defined by setting /(x) := x if xis rational, an...
- 4.1.15: Let f : :R ~ :R, let I be an open interval in :R, and let c e I. H ...
- 4.1.16: Let f : :R ~ :R, let J be a closed interval in R, and let c e J. If...
Solutions for Chapter 4.1: Limits of Functions
Full solutions for Introduction to Real Analysis | 3rd Edition
Addition property of equality
If u = v and w = z , then u + w = v + z
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable
Difference of complex numbers
(a + bi) - (c + di) = (a - c) + (b - d)i
See Power function.
Direction of an arrow
The angle the arrow makes with the positive x-axis
Endpoint of an interval
A real number that represents one “end” of an interval.
Theorem of Algebra A polynomial function of degree has n complex zeros (counting multiplicity).
An equation that is always true throughout its domain.
Instantaneous rate of change
See Derivative at x = a.
The instantaneous rate of change of a position function with respect to time, p. 737.
Length of a vector
See Magnitude of a vector.
Mode of a data set
The category or number that occurs most frequently in the set.
A logarithm with base e.
A function whose graph is symmetric about the origin (ƒ(-x) = -ƒ(x) for all x in the domain of f).
Data items more than 1.5 times the IQR below the first quartile or above the third quartile.
Product of a scalar and a vector
The product of scalar k and vector u = 8u1, u29 1or u = 8u1, u2, u392 is k.u = 8ku1, ku291or k # u = 8ku1, ku2, ku392,
A 90° angle.
Standard form: equation of a circle
(x - h)2 + (y - k2) = r 2
A number that is a sum of the arithmetic series 1 + 2 + 3 + ... + n for some natural number n.
The y-value of the bottom of the viewing window.
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