- 4.2.1: Apply Theorem 4.2.4 to determine the following limits: (a) lim (x +...
- 4.2.2: Determine the following limits and state which theorems are used in...
- 4.2.3: F. d lim JI + 2x - J1 + 3x h 0 . 1n 2 w ere x > . x-0 X +2x (b) lim...
- 4.2.4: Prove that lim cos(1/x) does not exist but that lim x cos(l/x) = 0....
- 4.2.5: Let 1, g be defined on A ~ R toR, and let c be a cluster point of A...
- 4.2.6: Use the definition of the limit to prove the first assertion in The...
- 4.2.7: Use the sequential formulation of the limit to prove Theorem 4.2.4(b).
- 4.2.8: Let n eN be such that n ~ 3. Derive the inequality -x2 !:: x 11 !::...
- 4.2.9: Let f, g be defined on A to 1R and let c be a cluster point of A. (...
- 4.2.10: Give examples of functions f and g such that f and g do not have li...
- 4.2.11: Determine whether the following limits exist in JR. (a) lim sin(ljx...
- 4.2.12: Let/: 1R-+ 1R be such that f(x + y) = f(x) + /(y) for all x, yin R....
- 4.2.13: Let A ~ 1R, let f: A -+ 1R and let c e 1R be a cluster point of A. ...
- 4.2.14: Let A ~ 1R, let f : A -+ R, and let c e R be a cluster point of A. ...
Solutions for Chapter 4.2: 4.2 Limit Theorems
Full solutions for Introduction to Real Analysis | 3rd Edition
Angle between vectors
The angle formed by two nonzero vectors sharing a common initial point
Completing the square
A method of adding a constant to an expression in order to form a perfect square
The function y = cot x
Difference of complex numbers
(a + bi) - (c + di) = (a - c) + (b - d)i
Difference of functions
(ƒ - g)(x) = ƒ(x) - g(x)
Direction vector for a line
A vector in the direction of a line in three-dimensional space
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
An identity involving a trigonometric function of 2u
A statement that compares two quantities using an inequality symbol
Using the science of statistics to make inferences about the parameters in a population from a sample.
Length of an arrow
See Magnitude of an arrow.
A procedure for finding the straight line that is the best fit for the data
See Numerical derivative of ƒ at x = a.
Recursively defined sequence
A sequence defined by giving the first term (or the first few terms) along with a procedure for finding the subsequent terms.
A square matrix with zero determinant
A function that maps real numbers to real numbers.
Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.
Union of two sets A and B
The set of all elements that belong to A or B or both.
A point that lies on both the graph and the y-axis.
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