Mathematics 242 – Principles of Analysis Problem Set 6 – due: Monday, March 24

’A’ Section

1. Determine whether each of the following limits exists using the “big theorem” for function limits and other results from section 3.2 of the text as needed.

(a) lim_{x→1} x^{2} − 5x + 3

(b) lim_{x→ \frac{1}{3}} x + \frac{1}{x^{2}}

(c) lim_{x→1} \frac{x^{3}−1}{x^{2}−1}

(d) Let

f(x) = \{ \begin{matrix}x^{1/3} sin (\frac{1}{x}) & if x ≠ 0 \\ 3 & 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{28}{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 and lim_{x→2} f(x) exists, 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. Show that your answers for parts a and d of 1 on the A section are correct using the ε, δ definition (not the “big theorem” or other results from Chapter 3, section 1 of the text.)

2. 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.)

3. 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.