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- Suppose we have the sequence of functions $f_n(x)=x^n$ defined on $[0,1],$ and suppose $f_n\to f$ pointwise where $f(x)=0$ if $x\in(0,1]$ and $f(x)=1$ if $x=1.$ Prove that Uniform Convergence fails.Suppose that fn : [0, 1] → R is defined by fn(x) = x n. If 0 ≤ x < 1, then xn → 0 as n → ∞, while if x = 1, then x n → 1 as n → ∞. So fn → f pointwise where Although each fn is continuous on [0, 1], their pointwise limit f is not (it is discontinuous at 1). Thus, pointwise convergence does not, in general, preserve continuity.Suppose x1 : 1/2 and xn+1 := x(n/2). Show that {xn} converges and find lim xn.
- Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.Let fn(x) = x/(n^2+x^2) for x ∈ R. Show that the sequence {fn} converges uniformly to the function that is everywhere zero.Let fn: R --> R be defined by : fn(x)= x/(1+nx2), For all n >= 1. a) Show that {fn} converges uniformly on R to a function f. b) Show that f'(x) = limn -->infinity f'n(x), For all x does not = 0, but this equality is false for x = 0. c)What assumption in the theorem on the interchange of the limit and thederivative is missing? I am stuck with that last part (C).
- Show whether or not fn(X) =sin (x/n) is pointwise convergenceA. Calculate the following limit, or explain why it does not converge. You may use any test to show that it doesn't converge.i. lim |x-2| / (x-2) (as x → 1+)(b) The series converges for every x in the half-open interval [−1, 1) but does not convergewhen x = 1. For a fixed x0 ∈ (−1, 1), explain how we can still use theWeierstrass M-Test to prove that f is continuous at x0.