Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?

1+ 2n² Xn {Tn} 1– 3n2 - Prove that the sequence where is bounded.

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.

Prove {(8n-5)/n} n=1 to infinity, is a Cauchy sequence in R with the usual metric dR(x,y) = abs(x-y)

Find the limit using l hopital rule.

Suppose that the sequence {n} is generated by the fixed point iteration method xn = g(xn-1), n ≥ 1. Further, suppose that the sequence convergence linearly to the fixed point p. Show that (a) g'(p) ~ _In-n-1 Xn-1 Xn-2 9 g' (p) (b) |en| ~₁ |In — Xn-1|- g' (p)-1