### Limits

As x approaches -2 from the left, what is the limit of (square root of x^2 +5) / (x+2) Please show work if you can. Choices are A. 3/2, B. 0, C. -infinity D. -1, E. + infinity

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As x approaches -2 from the left, what is the limit of (square root of x^2 +5) / (x+2) Please show work if you can. Choices are A. 3/2, B. 0, C. -infinity D. -1, E. + infinity

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lim [(cos x-1)x]/sin x x->0 Please see the attached file for the fully formatted problems.

(a) find the first 12 terms of the Fibonacci sequence Fn defined by the Fibonacci relationship Fn=Fn-1+Fn-2 where F1=1, F2=1. (b) Show that the ratio of successive F's appears to converge to a number satisfying r2=r+1. (c) Let r satisfy r2=r+1. Show that the sequence sn=Arn, where A is any constant, satisfies the Fi

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e. a) For a fixed integer n>0, let f(x) = (n+1)xn - nxn+1 . For x >1, show f is decreasing and that f(x) . Hence, for x >1; Xn(n+1-nx) < 1 b) Substitute the following x-value into the inequality from part (a)

Consider the series ∞ Σ ln (((k-1)(k+1))/k^2) = ln ((1 *3)/(2*2) + ln ((2*4)/(3*3)) +... k=2 a. Show the partial sum S4 = ln (5/8) b. Show the partial sum Sn = ((n+1)/(2n)) c. Use part b to show the partial sums Sn and therefore the series, converges.

Infinite Series : Convergence or Divergence of Geometric and P-Series and Integral Test (31 Problems) and Ratio and Root Test for Convergence (15 Problems)

Find the limit and justify your answer: lim n--> ∞ ∫ 0 --> ∞ sin nt/ (1 + t^2) dt Please see the attached file for the fully formatted problems.

Suppose that {an} and {bn} are sequences of positive terms, and that the limit as n goes to infinity of (an/bn) = L > 0. Prove that limit as n goes to infinity of an is positive infinity if and only if the limit as n goes to infinitiy of bn is positive infinity. Here is what I have for proving the first way: Suppose that

If |fn(x)| < gn(x) for all nE R and every x E[a,b] , and the series... converges uniformly in [a,b], then ... converges uniformly in [a,b]. Please see the attached file for the fully formatted problems.

Independence and relations Real Analysis Jacobians (VIII) Let u = (x + y)/(1 - xy) and v = tan inverse x + tan inverse y. If xy is not equal to 1,

Prove that the sequence of functions ... converges for every , and find the limit to which it converges. Please see the attached file for the fully formatted problems.

Find Limits and Rate of Convergence.

Suppose that f:X->Y is continuous.... --- (See attached file for full problem description)

------------------------------------------------------------------------------------------- 1). If g_n = Sup f_n, then prove that ( g_n)^-1 ( ( alpha, infinity] ) = union ( n = 1 to infinity) (f_n)^-1((alpha,infinity]). ------------------------------------------------------------------------------------------- 2). Pr

Prove that if lim(|c_(n+1)/c_n|) = a>0 then lim(|c_n|^1/n) = a

1). Find the radius of convergence for each of the following power series. Please check my solution for this problem: a). sum ( n = 0 to infinity) a^n z^n, a is a complex number. My solution: R( radius of convergence) = lim |a_n/a_n+1) = lim | a^n/a^(n+1)| = 1/|a| b). Sum ( n=0 to infinity) = lim|a^(n^2)*z^n, a is

1).If f: X--> C ( C is complex plane) is measurable, then prove that f^-1({0}) ( f inverse of 0 or any other point) is a measurable set in X. 2). If E is measurable set in X and if X_E ( x) = { 1 if x is in E, 0 if x is not in E} then X_E is a measurable function. Now I want you to prove the other direction, that is, I w

Please evaluate that attached limit... (See attached file for full problem description)

In the theory of relativity, the mass of a particle with velocity v is m = m0/sqrt(1- v^2/c^2) where m0 is the rest mass of the particle and c is the speed of light. What happens as v ?> c-?

Suppose X is a measurable space, E belongs to the sigma algebra ( I believe to the sigma algebra in X) , let us consider XE = Y. Show that all sets B which can be expressed as AE, where A belongs to the sigma algebra in X, form a sigma-algebra in Y. Please justify every step and claim you make in the solution.

Use the formal definition of the limit (epsilon-delta method) to show that: (See attached file for equation) Please be very thorough in your explanation of the solution by showing each step in detail. I really want to understand the method and be able to apply it to other problems. Thanks in advance! --- (See attach

Evaluate the following limit: lim (x-8) divided by, (the cubed root of x, minus 2) x->8

1) How would economic pressures like inflation or deflation affect your decision to make a long term investment? Should our mathematical analysis take these factors into consideration? 2) What are some different types of annuities that you have used or heard of?

Write f(z):=16z/(z^2 +1)^3 as f(z)= h(z)/(z - i)^3 with an explicit expression for the function h(z). Explain why h(z) has a Taylor series representation about i and use this representation to find explicitly the principal part of f at i. Hence, find the numerical value of the residue of f at i. Please see t

Please see the attached file for the fully formatted problems.

Find the limit if it converges if it does not diverge an = 1/n^2 + 2/n^2 + 3/n^2 + ...... n/n^2 does it diverge? If it converges whats the limit?

Please see the attached file for the fully formatted problems. 1) Prove that does not exist but that . 2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that . Prove that . 3) Let f, g be defined on A to and let c be a cluster point o

Determine if the summation series is convergent or divergent from n=2 to infinity of 1/n(ln(n))^2.