4. State the dominated convergence theorem and use it to find the limit when n→ ∞ of the following sequences : (a) f(t)dt, where ƒ is a continuous function on [0, 1], 1 (1+t") S (b) dt.

Q: Prove that if r >1 then f r" dx converges to 1

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Q: Let f : (a, b) → R. Prove that if f is differentiable at xo € (a, b), then the sequence {n(f(xo + )…

A: We will use the fact that f is differentiable at x_0 in (a,b) to show that the sequence converges to…

Q: Let X = C[0, 1] be a metric space with metric d(f,g) max {f(x) = g(x)}. x[0,1] Prove that the…

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Q: Suppose f is a continuous function on [0, ox) and that | f(t)dt converges to some positive real…

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Q: Consider the sequence of functions {fn}=2 where fn: [a, b] → R defined for n 2 2 by if 0<x s -n² (x…

A: Definition: Suppose fn is a sequence of functions fn:S→ℝ and f:S→ℝ. Then fn→f pointwise on S if…

Q: Let fn(x) on [0, 1]. 1+ nx (a) Does fn converge uniformly on [0, 1]? (b) Check if lim | fn(x) dæ lim…

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Q: -nx For n EN, consider the function f(x) = e on the interval [0, ∞). Let f(x) limn→∞ fn(x) for x =…

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Q: Assume that a > 0. For every n E N* and a E (0, 00, let fn(x) = n"re-n. (a) Show that the sequence…

A: We are given that for every n∈ℕ*and x∈[ 0 , ∞)  ,  fn (x) = nαx e-nx  , where  α&gt;0  To show :…

Q: Let fn(x) = n²x²(1 – x) on [0, 1]. On what closed subset, F C [0, 1], does the sequence {fn}…

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Q: #2

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Q: (a) State (without proof) the extreme value theorem. (b) Give an example of a function ƒ : [0, 1] →…

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Q: Consider a differentiable function f : R → R and assume that supx∈R |f′(x)| < 1. Show that there is…

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Q: Are the following statements true or false? If true give a proof, and if false give a…

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Q: Let f : (a,b) –R. Prove that if f is differentiable at xo € (a, b), then the sequence {n(f(xo + ±) –…

A: f:(a,b)→R is differentiable at x0∈(a,b)it means that     limh→0f(x0+h)-f(x0)h existwe replace h…

Q: divergence (D). IS TIOL (C), or (-1)"- 1)2 In(n + 1) (-1)"- 2) (In(n +1)) n = 1 1A, 2A b. a. e. 1C,…

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Q: 1. Consider a sequence of functions defined on R by giả th n f(z)= (i) Sketch the graphs of f1, f2.…

A: The given sequence of function is, fnx=x2+nxn. (i) To Sketch: graphs of f1,f2, f3, and f4 on same…

Q: 2. Using the sequential criteria for absence of uniform continuity, show that ƒ : R \ {0} → R given…

A: Letxn=1n;  yn=12n 1n-12n=2-12n=12n→0 as n→∞

Q: For n e N let fn: [0, 1] → R be the function fn(x) = e²-n. Decide whether the sequence (fn)converges…

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Q: Prove that f: R→ R is continuous. 2. Suppose that the function f: [0, 1] → R is continuous and that…

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