Assignment:
Q1. Prove that limx→0 cos(1/x) does not exist but that limx→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 limx→c g =0.
Prove that limx→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 limx→c ƒ and limx→c ƒ + g exist, then limx→c g exists.
(b) If limx→c ƒ and limx→c ƒ.g exist, does it follow that limx→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 limx→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.