1. Let f : (a, ∞) → and let k ∈ . Prove that limx→∞ k/f (x) = 0 whenever limx→∞ f (x) = ∞. 2. For each of the following conditions, find an example of functions f and g satisfying the condition with...

1. Let f : (a, ∞) →

and let k ∈
. Prove that lim_x→∞
k/f (x) = 0 whenever lim_x→∞
f (x) = ∞.

2. For each of the following conditions, find an example of functions f and g satisfying the condition with f → 0 and g → 0 as x → 0, but where lim_x→0+
f /g does not exist.

(a) f and g are nonzero in a deleted neighborhood of 0.

(b) lim_x→0−
f /g and lim_x→0+
f /g exist and are finite.

(c) g is nonzero in a deleted neighborhood of 0, but the one-sided limits in part (b) do not exist (finite or infinite)