MATH 295 PROBLEMS 11

IGOR KRIZ

1. Calculate

lim_{x→0} \frac{1 − \sqrt{cos x}}{x^{2}}.

[You can use

lim_{x→0} \frac{sin x}{x} = 1

without proof, but other than that try to work from first principles.]

2. Write

\frac{4x^{3} + 5x^{2} + 11x + 5}{x^{4} + 2x^{3} + 5x^{2} + 4x + 4}

as a sum of a polynomial, terms of the form

\frac{A}{(x − α)^{k}}

and terms of the form

\frac{Bx + C}{(x^{2} + bx + cx)^{k}} , b^{2} < 4c

(A, α, B, C, b, c real). [Hint: The polynomial x^{4} + 2x^{3} + 5x^{2} + 4x + 4 has root \frac{−1+i\sqrt{7}}{2}.]

3. Find complex numbers A, B, C such that

\frac{x}{(x − i)(x + i)^{2}} = \frac{A}{x − i} + \frac{B}{x + i} + \frac{C}{(x + i)^{2}} .

4. Describe the set of points (x, y) in the plane for which

sin(x + y) = sin x + sin y.

[Hint: \frac{sin x}{1−cos x} = cot(x/2).]

5. Does there exist a continuous function f : ℝ → ℝ whose graph intersects every non-vertical line in infinitely many points?