Suppose that a = (a1 an) Rn, that fj: R R for

Question:

Suppose that a = (a1 ∙ ∙ ∙ an) ∈ Rn, that fj: R → R for j = 1, 2,..., n, and that g(x1, x2,..., xn) := f1(x1) ∙ ∙ ∙ fn(xn).
a) Prove that if fj(t) fj(a) as t → aj, for each j = 1,..., n, then g(x) → f1(a1)∙ ∙ ∙ fn(an) as x → a.
b) Show that the limit of g might not exist if, even for one j, the hypothesis "fj(t) → fj(aj)" is replaced by "fj(t) → Lj for some Lj ∈ R.