Let {ƒn (x)}~_₁ be a sequence of functions. ptwise Explain very clearly the difference between ƒn (x) →→→→→ ƒ(x) and ƒʼn (x) unif → ƒ(x).
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- Let fn(x) = nx/1 + nx2 (a) Find the pointwise limit of (fn) for all x ∈ (0,∞).Let x1 > 1 and xn+1 = 2 − 1 / xn for n ≥ 2. Show that ( xn ) converges and find its limit.Find the linearization of f(x)=ln(x2-3) at suitably chosen integer near X=2.1. Then use the linearization to estimate the value of f(2.1).
- Suppose that F(u) denotes the DFT of the sequence of f(x)={1, 2, 3, 4}? What is the value of F(14)? (Hint: DFT periodicity)(b) A sequence (fn) of differentiable functions such that (fn ) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.Find the absolute maximum and absolute minimumvalues of f on the given interval. f(x) =12 +4x - x2 , [0, 5]
- In Example 2.4.1, show that fn(x) converges to f(x) pointwise on [0, 1]. (We haveseen the convergence at x = 0, 1.) For x ∈ [0, 1] and positive integer n, let fn(x) = xn.Then limn→∞ fn(x) = f(x),, wheref(x) = 0 if 0 ≤ x < 1,1 if x = 1.This example shows that the pointwise limit of a sequence of continuous functions need not be continuous.Find the absolute maximum and absolute minimum of the function f ( x ) = x 2 e -x on the closed interval [ 1 , 4 ].Find x so that x, x + 2, and x + 3 are consecutive terms of ageometric sequence.