1. Suppose that f (x) = g (1/x) for x > 0 and let L ∈ R. Prove that limx→ f (x) = L iff limx→0 g (x) = L. 2. Let f : (a, ) → . Prove that limx→ f (x) = L iff for every sequence (xn) in (a, ) with...

1. Suppose that f (x) = g (1/x) for x > 0 and let L ∈ R. Prove that lim_x→

f (x) = L iff lim_x→0
g (x) = L.

2. Let f : (a,
) →
. Prove that lim_x→


f (x) = L iff for every sequence (x_n) in (a,
) with lim_n→

x_n
=
, the sequence ( f (x_n)) converges to L

3. Let f : (a,
) →
. Prove: If the limit of f as x →

exists, then it is unique

4. Let f and g be real-valued functions defined on (a,
). Suppose that lim_x→

f (x) = L and lim_x→

g (x) = M, where L, M ∈
. Prove the following.

(a) lim_x→

( f + g)(x) = L + M

(b) lim_x→

( fg)(x) = LM

(c) If k ∈
, then lim_x→∞
(k f )(x) = k L.

(d) If g (x) ≠ 0 for x > a and M ≠ 0, then lim_x→

( f /g)(x) = L/M.