Mathematics 242 – Principles of Analysis Problem Set 6 – due: Friday, March 22

’A’ Section

1. Determine whether each of the following limits exists, then prove that your answers are correct using the ε, δ definition.

(a) lim_{x→3} x^{2} − 4x + 1

(b) lim_{x→ \frac{1}{2}} x + \frac{1}{x}

(c) lim_{x→2} \frac{x^{3}−8}{x^{2}−4}

(d) Let

f(x) = \{ \begin{matrix}x sin (\frac{1}{x}) & if x ≠ 0 \\ 2 & if x = 0 \end{matrix}

and consider lim_{x→0} f(x).

2. Which of the functions in question 1 are continuous at the indicated c in the limits there? Explain.

3. True-False. For the true statements, give a short proof. For the false statements give a counterexample.

(a) If lim_{x→1} f(x) = e − \frac{27}{10}, then there exists a δ > 0 such that f(x) > 0 for all x with 0 < |x − 1| < δ.

(b) If |f(x)| ≤ x^{3} for all x, then lim_{x→2} f(x) = 8.

(c) Let f : R → R be defined by this rule:

f(x) = \{ \begin{matrix}2x & if x is rational \\ −2x & if x is irrational. \end{matrix}

Then lim_{x→0} f(x) exists and equals 0.

(d) If f(x) < g(x) on a deleted neighborhood of c, lim_{x→c} f(x) = L, and lim_{x→c} g(x) = M, then L < M.

’B’ Section

1. Assume that lim_{x→c} f(x) = L.

(a) Show that there exists a constant B and δ > 0 such that |f(x)| ≤ B for all x in the deleted neighborhood {x ∈ R | 0 < |x − c| < δ}.

(b) Using part (a), not the limit product rule, show that lim_{x→c}(f(x))^{n} = L^{n} for all integers n ≥ 1.

(c) Assume that f(x) ≥ 0 on some deleted neighborhood of x = c. Show that

lim_{x→c} \sqrt{f(x)} = \sqrt{L}.

(Hint: It may help to treat the cases L = 0 and L ≠ 0 separately.)

2. In this problem you will show that

lim_{θ→0} \frac{sin(θ)}{θ} = 1.

For 0 < θ < \frac{π}{2} , the point P = (cos(θ), sin(θ)) = (x, y) lies on the arc of the unit circle x^{2} + y^{2} = 1 in the first quadrant.

(a) Let O = (0, 0), Q = (cos(θ), 0), and R = (1, 0). (Draw a picture!) By considering the areas of the triangle ΔOQP and the circular sector ORP, deduce that if 0 < θ < \frac{π}{2} , then sin(θ) cos(θ) ≤ θ. (You may use “intuitively reasonable” facts about areas such as the statement that if one plane region R is completely completely contained in a second region S, then area(R) ≤ area(S).)

(b) Now take the tangent line to the circle at R (a vertical line), and let S = (1, tan(θ)) be the intersection of that line and the radius OP (extended). Considering the areas of the triangle ΔORS and the sector ORP as above, explain why θ ≤ tan(θ).

(c) Combine parts (a) and (b) to deduce that if 0 < θ < \frac{π}{2} , then

cos(θ) ≤ \frac{sin(θ)}{θ} ≤ \frac{1}{cos(θ)}.

(d) Using the one-sided form of Theorem 3.2.9 (The Limit Squeeze Theorem), show that

lim_{θ→0^{+}} \frac{sin(θ)}{θ} = 1.

(You will need to use the fact that cos(θ) is continuous at θ = 0.)

(e) Now, for −\frac{π}{2} < θ < 0, show that \frac{sin(θ)}{θ} = \frac{sin(|θ|)}{|θ|} and use this to see that

lim_{θ→0^{−}} \frac{sin(θ)}{θ} = 1

as well.

(f) Finally, explain how parts (d) and (e) combine to show the statement at the start of the problem.