- 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
In an experiment with two possible outcomes, the probability of one outcome occurring k times in n independent trials is P1E2 = n!k!1n - k2!pk11 - p) n-k where p is the probability of the outcome occurring once
See Compound fraction.
A coordinate plane used to represent the complex numbers. The x-axis of the complex plane is called the real axis and the y-axis is the imaginary axis
The function y = cos x
The gathering and processing of numerical information
See Polar coordinates.
Equivalent equations (inequalities)
Equations (inequalities) that have the same solutions.
The area of ¢ABC with semiperimeter s is given by 2s1s - a21s - b21s - c2.
The domain of a function’s algebraic expression.
Magnitude of a real number
See Absolute value of a real number
Multiplication principle of probability
If A and B are independent events, then P(A and B) = P(A) # P(B). If Adepends on B, then P(A and B) = P(A|B) # P(B)
Two lines that are both vertical or have equal slopes.
The measure of acidity
A degree 4 polynomial function.
Sum of two vectors
<u1, u2> + <v1, v2> = <u1 + v1, u2 + v2> <u1 + v1, u2 + v2, u3 + v3>
p = ƒ(x), where x represents production and p represents price
The function y = tan x
A circle with radius 1 centered at the origin.
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.