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# Real Analysis

### Convergent Power Series and Heat Equation

Show that there always exist a convergent power series solution to the heat equation with u(x,0)=p(x)=polynomial. Is the solution a polynomial?

### Convergence, Divergence and Sums of Series

Could you please check if the answers are right? Please see the attached file for the fully formatted problems.

### Trigonometric Limits

Use one-sided limits to find the limit or determine that the limit does not exist. 16-x^2 /4-x lim x => 4 Find the trigonometric limit: sin3x/2x limx => 0 Please show work.

### Antiderivatives General Form

? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of . ? Find the most general form of the antiderivative of Please see the attached file for the fully formatted problems.

### Finding Limits

Please see the attached file for the fully formatted problems. Let be defined as a function such that where and where is defined as I would be grateful for a proof that the limits of are and I think that l'Hopital rule is useful here.

### Difference Quotient : Limits and Differentiable Functions

Assume f:(-1,1) --> R and f'(0) exists. If a_n , b_n -> 0 as n->infinity, define the difference quotient: D_n = ( f(b_n) - f(a_n) ) / ( b_n - a_n). a) Prove lim [n -> infinity] D_n = f'(0) under each condition below: (i) a_n < 0 < b_n . (ii) 0 < a_n < b_n and (b_n) / (b_n - a_n) <= M (iii) f'(x) exists and is contin

### Real Analysis using the Lipschitz Condition

Please see the attached file for the fully formatted problems.

### Real Analysis: Limits Proof

Prove sqrt(x+1) - sqrt(x) goes to 0 as x --> infinity Please see the attached file for the fully formatted problems.

### Real Analysis : Hilbert Space, Inner Product and Maps

Let V=L2[-1,1] be the Hilbert space of functions over the time interval [-1,1] with inner product ...... Let P 5 V be the subspace of polynomials of order 4 or less, endowed with the inner product and norm of V, and let ..... be its natural basis. Define a linear transformation S as ...... Show that the subspace P 5 is

### Measure Theory and Dominated Convergence Theorem

Please see the attached file for the fully formatted problems. I have provided a solution to the attached problem. I do not understand or like the solution - I was hoping you could provide an alternate solution or expand upon the solution I have provided in more detail. Exercise (moment-generating function). ? Let X be

### Taylor's Series for a function of two variables

Find the Taylor's series expansion of f(x, y) = sin (e^y + x^2 - 2) around(1, 0).

### Taylor's series of two variables.

Find the Taylor's Series expansion of f(x, y) = sin x sin y about (0, 0).

### Finding Limits by Graphing

Please see attached file. Graph appropriate functions and then do the limits.

### Finding Limits from a Table of Values

Estimate the value of the limit by filling out a table and then making an educated guess about the tendency of the numbers. Then graph the lim of tanx///x as x approaches 0 to verify own conclusion lim of tanx///x as x approaches o table should be 2 colums with the headings |x|tanx/x|

### Function of Limits

The function has a limit as f(x) = (1/x) + 3 has a limit of L=3 as x approaches x. This means that if x is sufficiently large (that is if x > N for some number N), the values of f(x) are closer to L=3 than a number epsilon > 0. a) Sketch the graph y=(1/x) +3 and a horizontal strip of points (x,y) such that (if y

### Finding Limits Square Roots

Find the limit lim of square root of12x^3-4x+1/1+3x+2x^3 as x approaches + infinity

### Limits : Horizontal and Vertical Asymptotes

Find the horizontal and vertical asymptotes of A) f(x) - x^2 - 3/x^2+1 B) f(x) = x-1/x^2-5

### Finding Algebraic Limits

Use algebra and limits to find the limits: a) lim of x+1/x-2 as x approaches +infinity b) lim of x^2+1/x^2 - 5 as x approaches -infinity c) lim of x^2+1/x^3 - 2 as x approaches + infinity d) lim of x^3+1/x^2 + 2 as x approaches - infinity check answers by graphing the functions

### Evaluating Limits by Graphing

Graph the appropriate functions (first try by hand and then verify your sketches with a grapher and then determine the limits. a) lim of lx as x approaches 0+ b) lim of 1/x as x approaches 0+ c)lim of ln 1/x as x approaches 0- d)lim of 1/x as x approaches 0 e)lim of 1/ x^2 as x approaches 0+ f)lim of -1/ x^2 as x approac

### Limits and Continuous Functions

Sketch by hand the graph of F that satisfies: (a) f(0) is not defined, (b) lim as x approaches 0 from left of f(x) does not exist and (c) lim as x approaches 0 from right f(x)=2. is the function f(x)continuous from the left at x = 0? Is the function f(x) continuous at x=0 from the right? is the function f(x) continuous at x = 0

### Evaluating Limits

Use limit laws to find the limits lim as x approaches 7 lim of square root (x) - square root (7)/x-7

### Evaluating Limits as they Approach Values

Find the limits: a)lim as x approaches zero from left of 2x b)lim as x approaches 3 from righ of 2x of x c)lim as x approaches 3 from right of 5 d)lim as x approaches -1 from left of|x-2|

### Description of Evaluating Limits

Use the limit laws to find the limits: lim as x approaches 3of x^2-x+2

### Limits

Find lim as x approaches-1 from the left of f(X). find lim as x approaches-1 from the right of f(X). Are these limits the same? What can you say about find lim as x approaches 1 of f(X)?

### Investigating Limits

Estimate the value of the lim as x approaches zero of cosx/x, if it exists, by filling out a table and than making an educated guess about thr tendency of the numbers. The first row in the table should start like this: X|cosx|x, with approximately 6 more values added for both x and cox/x Then graph y=cos/x to verify your

### Limits: Graphing Functions

Use the graph of the function f(x)=1/(1+e^(1/x)) to discuss the lim as x goes to zero f(x).

### Find the Limits

What is the limit as x approaches infinity of 4x^2 / (2x^3 - 3x + 6)? Is it 0? What is the limit as x approaches -4 of (x^2 -16)/(x+4)? I got -8.

### Real analysis problem rational cuts in Q

A pdf of the problem is attached. See attached file for full problem description.

### Real analysis problem with dedekind cuts

A pdf document of the problem is attached.

### Real analysis question with collection of subsets

I have a problem deal with the subject of real analysis and it is about the collection of subsets. I hope someone can help me with detail explanation. See attached file for full problem description.