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# Real Analysis

### Real Analysis : Countability

1. Show that the set of infinite sequences from is not countable. Hint: Let be a function from to . Then is a sequence . Let . Then is again a sequence from , and for each we have . This method of proof is known as the Cantor diagonal process. 2. Show that is uncountable. (Use Problem 1) Please see the

### Limits and Continuity

Find Lim g(x) x>1+ , Lim g(x) x>1- , Lim g(x) x>1 Find g(1) Is g continuous at x=1? Why, why not? Find Lim x>-2 Find g(-2) Is g continuous at x= -2? Why, Why not?

### Analysis: Finding a Limit

Find limit x>> -1^+ f(x) f(x)= x - 2, for x <= 3; x - 1, for x > 3 Also: When you first begin to draw the graph of f(x), why do you start where you do? How high do I go when drawing a graph? How do I know when to stop?

### Limit

Lim (2x^3 - 7x^2 + 3x - 4)/(3x^3 + 5x - 6) = x=> oo 2/3 3/2 0 1 -1 oo none of these

### Anti-Derivatives

Please choose the correct answer 19. [g(x)]^-1g'(x) dx = (-1/2)[g(x)]^(-2) + C -[g(x)]^(-2) + C ln |g(x)| + C -1/[g(x)] + C none of these 20. True or False: x^14/14 + sqrt(13) + e^x is an antiderivative of x^13 + e^x True False.

### Find Functions and Evaluate Limits

1. Find functions f and g such that f(x) = (f.g)(x) where f(x) = (x^2+1)^0.205 2. Evaluate lim lim f(x) = -x^2+1 x< = -1 x-> -1 f(x) = 2x^2+3x x > -1.

### Finding and Evaluating Limits

Please find the limits and justify the answer: a) lim 2^x - x² + pi x‾0 ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ x^10 b) lim sin 2x x‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ c) lim (1 - 1/x)^x

### Real Analysis: Find the Limit

Find the limit lim 3e^2x - 3 x→0 ¯¯¯¯x¯¯¯¯ by recognizing it as a derivative f'(a) of the appropriate function at a suitable value of a. Please specify the function f in the answer - numerical computation of the limit is not enough. (^ means exponent and 3e^2x - 3 is over x).

### Problem: Finding Limits

Find the limit lim cos x x&#8594;&#8734; ¯¯x¯¯¯ (cos x is over x - I am not able to make a continuous line)

### Limit: L'Hopital's Rule

Please explain the steps and solutions, thanks: Evaluate the limit: lim (1+ x)^1/x x&#8594;&#8734; (^ means exponent and 1/x is the exponent)

### Evaluating Limits with Exponents

Evaluate the limit: lim (e^x + x)^1/x x→0 (^ means exponent and 1/x the exponent)

### Limit

Evaluate the limit: lim x cot x x&#8594;0

### Finding Limits using L'Hopital Rule

Find the limits and justify the answers: a) lim &#8730;xlnx x&#8594;0^+ b) lim (1+2x)^1/x x&#8594;0^+ (^ means exponent and ^1/x is the exponent 1 over x)

### Algebra Analysis: Limits

A) lim 3x x-->0 overlineoverlineoverlineoverlineoverlineoverlineoverline = 3 x^2+x b) lim x does not exist because one cannot divide by 0 x-->0 overlineoverlineoverlineoverlineoverlineoverline x^2+x c) If f (5) is n

### Evaluating Functions and Limits

H is a function such that h(0) = 1, h(2) = 7, h(4) = 5, h'(0) = -2, h'(2) = 3, and h'(4) = -1 Evaluate lim h(w)&#8722;1 w&#8594;0 ¯¯¯w¯¯¯

### Evaluating Limits (7 Problems)

Please see the attached file for the fully formatted problems.

These problems may or may not need PHSTAT again. All is the same as priors posting the responses in a MS Word document. Attached are: -Problems [MS Word] -Spreadsheet [Excel]

### Taylor Series Expansion Process

Find the Taylor series expansion for the function for the given value of a. F(x) = √ x, a= 4

### Maclaurin Series Expansion Nonzero Terms

Evaluate the Maclaurin series expansion using the first three nonzero terms: 1 &#8747; (cos x - 1)/x dx 0 keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple

### Maclaurin Series Expansion Function

F(x) = sin 4x

### Sums of Convergent Series

How many terms of the convergent series do you need in order for the partial sum Sn to estimate the actual sum of the series correctly to four decimal places? ( The sum of the series from n=1 to infinity of (-1) ^5n divided by Ln n.)

### Some problems on network models

I have attached some quantitative Analysis Questions that I need help solving so I can add to my study guide. The book used was Quantitative analysis for Management by Barry Render but as you already know these concepts are found in any Quantitative Analysis book.

### Decision Analysis : Influence Diagrams, Decision Trees and Optimistic, Conservative, and Minimax Regret Approaches

3. Southland Corporation's decision to produce a new line of recreational products resulted in the need to construct either a small plant or a large plant. The selection of plant size depends on how the marketplace reacts to the new product line. To conduct an analysis, marketing has decided to view the possible long-run dema

### Terms of Power Series

4. Let f be the function by f(x)e^(-2x^2) a. Find the fist four nonzero terms and the general term of the power series for f(x) about x = 0. b. Find the interval of convergence of the power series for f(x) about x =0. Show the analysis that leads to our conclusion. c. .... Please see the attached file for the fully formatted

### Convergence and Divergence of Series

Page Number: - 568 1 / (b, e, g, h) Page Number :- 576 1 / (c, d), 5 / ( a, c, d, f) Page Number: - 579 1 /( e , f, h), 3 Please explain all steps.

### Spherical Coordinates, Continuous Functions, Polar Coordinates and Limits

Determine the set of points at which the function is continuous. 27. F(x, y) = sin(xy)/(e^x - y^2) Use polar coordinates to find the limit. [If (r, theta) are polar coordinates of the point (x, y) with r >= 0, note that r --> 0+ as (x, y) --> (0,0).] lim (as (x,y) --> (9,0)) of (x^3 + y^3)/(x^2 + y^2) Use spherical

### Remainder Estimation Theorem and Euler's Formula

1.) Given that 1-x+x^2+...+(-x)^n is a power series representation for 1/(1+x), find a power series representation for (x^3)/(1+x^2). 2.) The Maclaurin series for f(x) is: 1+2x+((3x^2)/(2))+((4x^3)/(6))+...+(((n+1)x^n)/(n!))+... Let h(x)= (the integral from 0 to x) f(t) dt. Write the Maclaurin series for h(x).

### Sum of a Series

1- prove : 1/2^2 + 1/3^2 +1/4^2+ ..... + 1/n^2 < 1

### Taylor Series and Systems of Vector Equations

1 Write the Taylor series with center zero for the function f(x) = ln(l+x^2). 2 Given a =2i +3j, b = 3i +5j , and c = 8i + 11j express c in the form ra+ sb where r and s are scalars.

### Real Analysis: Set Measures and Measurability

9. Let F be a closed subset in R, and ... the distance from x to F, that is, ....... Clearly, ....) whenever .... Prove the more refined estimate .... for a.e. xEF, that is,..... [Hint: Assume that x is a point of density of F.] Please see the attached file for the fully formatted problems.