Find an example of function f g r - 0 rarr r such that

Question

Exercise 1 - Find an example of function f, g: R - {0} → R such that lim_x→0f(x) = 0, and that limx→0[gf](x) does not exist.

Exercise 2 - Let I ⊆ R be an open interval, let c ∈ I and let f: I - {c} → R be a function.

(1) Let L ∈ R. Prove that if lim_x→cf(x) = L, then lim_x→c|f(x)|=|L|.   

Theorem - Let I ⊆ R be an open interval, let c ∈ I and let f, g: I - {c} → R be functions. Suppose that lim_x→cf(x) exists and limx→cf(x) ≠ 0, that g(x) ≠ 0 for all x ∈ I - {c} and lim_x→cg(x) = 0.

1. lim_x→c[f/g](x) does not exist.      

2. If lim_x→cf(x) > 0 and g(x) > 0 for all x ∈ I - {c}, or if lim_x→cf(x) < 0 and g(x) < 0 for all x ∈ I - {c}, then lim_x→c[f/g](x) = ∞. If lim_x→cf(x) > 0 and g(x) < 0 for all x ∈ I - {c}, or if lim_x→cf(x) < 0 and g(x) > 0 for all x ∈ I - {c}, then lim_x→c[f/g](x) = -∞.