- 4.3.1: Prove Theorem 4.3.2.
- 4.3.2: Give an example of a function that has a right-hand limit but not a...
- 4.3.3: Let f (x) := lxl-112 for x #:. 0. Show that lim f(x) = lim f (x) = ...
- 4.3.4: Let c e 1R and let f be defined for x e (c, oo) and f(x) > 0 for al...
- 4.3.5: Evaluate the following limits, or show that they do not exist. (a) ...
- 4.3.6: Prove Theorem 4.3.11.
- 4.3.7: Suppose that f and g have limits in R as x --+ oo and that f (x) !:...
- 4.3.8: Let f be defined on (0, oo) toR. Prove that lim f(x) = L if and onl...
- 4.3.9: Show that iff: (a, oo)--+ R is such that lim xf(x) = L where L e R,...
- 4.3.10: Prove Theorem 4.3.14.
- 4.3.11: Suppose that lim /(x) = L where L > 0, and that lim g(x) = oo. Show...
- 4.3.12: Find functions f and g defined on (0, oo) such that lim f = oo and ...
- 4.3.13: Let f and g be defined on (a, oo) and suppose lim f = L and lim g =...
Solutions for Chapter 4.3: Some Extensions of the Limit Conceptt
Full solutions for Introduction to Real Analysis | 3rd Edition
Additive inverse of a complex number
The opposite of a + bi, or -a - bi
Union of two rays with a common endpoint (the vertex). The beginning ray (the initial side) can be rotated about its endpoint to obtain the final position (the terminal side)
See Arithmetic sequence.
Difference of functions
(ƒ - g)(x) = ƒ(x) - g(x)
For the equation ax 2 + bx + c, the expression b2 - 4ac; for the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the expression B2 - 4AC
The minimum, first quartile, median, third quartile, and maximum of a data set.
Reciprocal of the period of a sinusoid.
Graph of an equation in x and y
The set of all points in the coordinate plane corresponding to the pairs x, y that are solutions of the equation.
The graph of the linear inequality y ? ax + b, y > ax + b y ? ax + b, or y < ax + b.
Side opposite the right angle in a right triangle.
Left-hand limit of f at x a
The limit of ƒ as x approaches a from the left.
A distribution of data shaped like the normal curve.
The graph of parametric equations.
The numbers (r, ?) that determine a point’s location in a polar coordinate system. The number r is the directed distance and ? is the directed angle
The first quartile is the median of the lower half of a set of data, the second quartile is the median, and the third quartile is the median of the upper half of the data.
Rational zeros theorem
A procedure for finding the possible rational zeros of a polynomial.
Solution of an equation or inequality
A value of the variable (or values of the variables) for which the equation or inequality is true
Standard form of a complex number
a + bi, where a and b are real numbers