### Limits

Please see the attached file for the fully formatted problem. Lim (1 - cos t)/2t t--> 0

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Please see the attached file for the fully formatted problem. Lim (1 - cos t)/2t t--> 0

See attached

Evaluate Limit[tan (a*theta)/sin(b*theta)] as theta approaches zero.

Calculate limit (1/x^2) as x approaches infinity

Prove that Lim (3x)=6 as x approaches 2 using Epsilon-Delta definition.

Define a limit graphically

Determine whether the series Sum(n!/n^n) n=1..infinity is absolutely convergent, conditionally convergent or divergent.

Find lim (ln x)^3/(x) as x--> infinity

Test the series (in the attached file) for convergence or divergence by using the Comparison Test or the Limit Comparison Test.

Please see the attached file for the fully formatted problem. What is the maximum of F = x1 +x2 +x3 +x4 on the intersection of x21 +x22 +x23 + x24 = 1 and x31+ x32+ x33+ x34= 0? As this is an analysis question, please be sure to be rigorous and as detailed as possible.

The question is in attached file. Suppose a sequence of continuous functions, { ?n }, has the property that ?n  ? and  > 0,   > 0 such that if | x - y | <  then n, | ?n (x) - ?n (y)| <  Prove that ? is continuous.

Questions are in attached files. Thank you.

Summation Series. See attached file for full problem description.

I have managed to evaluate the double integral using the horizontal simple method, and answer 63. But when I reverse the order (vertical simple method) I cannot reach the same answer of 63, I get 65. 1) Evaluate the double integral of f(x,y)=x+4y^2 over the triangular region with vertices (-2,2) (4,2) & (1,-1) Check that r

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 t

I need a proof for "If f on [a,b] is continuous & 0 is not a member f([a,b]) then f is bounded away from 0."

For numbers a1,....,an, define p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that: (a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0 Prove that there is some point x in the interval (0,1) such that p(x) = 0

Let f:[a,b] mapped to the Reals be a function that is integrable over [a,b], and let g:[a,b] mapped to the Reals be a function that agrees with f except at two points. Prove g is integrable over [a,b].

Prove that if f : [a,b] ----> R is a bounded function that is continuous at all but finitely many points, then f is integrable over [a,b].

The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is.

Suppose that the function F:R->R has derivatives of all orders and that: F"(x) - F'(x) - F(x) = 0 for all x F(0)=1 and F'(0)=1 Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.

Please see the attached file for the fully formatted problems. Suppose ak 0 and a1/k k ! a as k ! 1. Prove that P 1k =1 akxk converges absolutely for all |x| < 1/a if a 6= 0 and for all x 2 R if a = 0. Since this problem is an analysis problem, please be sure to be rigorous.

Please see the attached file for the fully formatted problem. Define ak recursively by a1 = 1 and ak = (−1)k 1 + k sin 1 k −1 ak−1, k > 1. Prove that P 1k =1 ak converges absolutely. Since this problem is an analysis problem, please be sure to be rigorous.

Please see the attached file for the fully formatted problems. Let f be a real function defined by . 1) Evaluate f'(x), f''(x), f(0). Show that f has exactly two roots and , with . Find an interval of two consecutive real numbers within which the roots must lie. From now on, let us denote and these two (closed) in

Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) N

Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation

Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c<d<=b Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

Define f(x) = x^2 for all x in [0,1]. For each natural number n, compute L(f,Pn) and U(f,Pn), where Pn is the regular partition of [0,1] into n subintervals.Then use the Integrability Criterion to show that the function f:[0,1]->R is integrable.

Please see the attached file for the fully formatted problems. Prove that if f is integrable on [0, 1], then lim n !1 Z 1 0 x n f(x)dx = 0 Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition,

Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f