MATH 295 PROBLEMS 6

IGOR KRIZ

1. Let f(x) be a real function of one variable whose domain contains [a, +∞) for some a. Write down a definition of

lim_{x→+∞} f(x) = +∞.

2. Compute from first principles

lim_{x→∞} \sqrt{x^{2} + 2x} − x.

[Hint: Use theorems about behaviour of limits under arithmetic operations and composition.]

3. Prove that if

lim_{x→a} f(x) = ℓ,

then there is a number δ > 0 and a number M such that |f(x)| < M for 0 < |x − a| < δ.

4. Prove that if

lim_{x→0} f(x)/x = ℓ

and b ≠ 0, then

lim_{x→0} f(bx)/x = bℓ.

5. Define a function f : (−∞, 0) ∪ (0, +∞) → ℝ by

f(x) = 1/n

where n is the smallest positive integer satisfying 1/n < |x|. Prove that

lim_{x→0} f(x) = 0.