Real Analysis

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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.

? 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

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