Limits and cluster points

Question

Assignment:

Q1. Prove that lim_x→0 cos(1/x)  does not exist but that lim_x→0 cos(1/x) = 0.

Q2. Let f, g be defined on A ⊆ ℜ to ℜ , and let c be a cluster point of A.  Suppose that f is bounded on a neighborhood of c and that lim_x→c g =0.

Prove that  lim_x→c ƒ.g =0.    .

Q3. Let f, g be defined on A to ℜ and let c be a cluster point of A.
    
(a) Show that if both lim_x→c ƒ and lim_x→c ƒ + g exist, then lim_x→c g exists.
(b) If lim_x→c ƒ and lim_x→c ƒ.g  exist, does it follow that lim_x→c g exists?

Q4. Let f : ℜ→ℜ be such that f(x+y)=f(x)+f(y) for all x, y in ℜ .  Assume that lim_x→0 ƒ exists.  Prove that L = 0, and then prove that f has a limit at every point c ∈ ℜ.  (Hint: First note that f(2x) = f(x)+f(x) = 2f(x) for x∈ ℜ  .  Also note that f(x) = f(x – c) + f(c) for x, c in ℜ )

Provide complete and step by step solution for the question and show calculations and use formulas.