MATH 295 PROBLEMS 7

IGOR KRIZ

1. Calculate:

lim_{x→+∞}(x − \sqrt[3]{x^{3} − 1}).

2. Let (a_{n}) be a sequence and let f be a real function of one variable whose domain contains {a_{n}|n ∈ ℕ}. Suppose that

lim_{n→∞} a_{n} = a, lim_{x→a} f(x) = b.

Prove that

lim_{n→∞} f(a_{n}) = b,

provided that a_{n} ≠ a for all but finitely many n, or f(a) = b. [Hint: Mimic the proof of the theorem about the limit of a composition of functions.]

3. Suppose that a function f : ℝ → ℝ satisfies f(x + y) = f(x) + f(y) for all x, y, and that f is continuous at 0. Prove that f is continuous at all a. [A function f is continuous at a is

lim_{x→a} f(x) = f(a).]

4. Give an example to show that the following definition of

lim_{x→a} f(x) = ℓ

is not correct:

For all ϵ > 0, there exists a δ > 0 such that if |f(x)−ℓ| < ϵ, then 0 < |x−a| < δ.

5. Prove that, for a_{1}, ..., a_{n} > 0,

\frac{n}{\frac{1}{a_{1}} + ... + \frac{1}{a_{n}}} ≤ \sqrt[n]{a_{1}. ... .a_{n}}.

The left hand side is called the harmonic mean of a_{1}, ..., a_{n}. [Hint: Use the inequality between arithmetic and geometric mean.]