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: Q4: Consider the sequence (f, (x)) of functions f,:R→R defined by : nx f,(x) = Ji+n°x² Find the…

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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: Show that if (Sntn) converges to (s,t) in R² with the do ("max") metric, then the sequence converges…

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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: Let f: R→ R be a function such that f(x) ≥ x, for all x E R. Show that if Ef (bn) converge, then Σb,…

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Q: Q 5. For -∞0 < x <∞, define -{ Show that f is not continuous at z = 0. [Hint: Use sequences.] f(x) =…

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Q: #2

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Q: a. 0 a.c .Q. Consider (R, M(R), µ). For x € [2,3], consider the sequence of functions tan ((x –…

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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: Let fn (x) uniformly converge to f(x) on [a, b]. If f„(x) E R[a, b], then f(x) E R[a, b] O True…

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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: For each of the following sequences, find the z-transform of the sequence, sketch the pole-zero…

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Q: 7. Prove that for every x E (-1,1) 1 1– 2x + 3x2 – 4x³ E(-1)" (n + 1)x" : - (1+x)² ` n=0 Make sure…

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Q: a) Let (h,). (t) be sequences of hounded functions on A that converges uniformly on A to h,t…

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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: Q5. Given f(x): (R.) → (R₁) 3x+2 if x 22 5x-1 if x < 2 f(x) = { (a) Let {n} = {2} be a sequence in…

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Q: Consider the sequence of functions {fn}nen, where nx fn(z) = Va € [1, 10]. 1+ nx' (a) Find the…

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Q: Let a be increasing in [a, b]. Suppose a sequence of real continuous functions in [a, b] converges…

A: Given that α in increasing in a,b and a sequence of real continuous function fn in a,b converges…

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: (c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all a E R. Show that the…

A: Please check step 2 for the solution.!

Q: a] If the function f(x) is continuous in a closed interval [a,b] and f(a)f(b)<0, Prove that the…

A: let us prove the first part by considering a function let fx=x, x not equal to zero  and f0=1 but as…

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: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: Let f : A → R and (xn),(Yn) are two sequences converges to c such that Xn, Yn # c, Vn E N. If…

A: We understand this statment by the help of an counterexample .

Q: Find an example of a sequence of continuous functions {fn} defined on the interval [0, 1] such that…

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Q: 3 (c) Apply the monotone convergence theorem to evaluate Lim fn(x)dµ.

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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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