Unless otherwise stated, all functions are assumed to map the real numbers into the real numbers. The letter, R, denotes the set of real numbers.

1. Let the function be defined by: f(x) = x if x is rational, 0 if x is irrational
Show that limₓ→₄ does not exist and therefore that f(x) is not continuous at x = 4.

2. Write the definition of continuity at x = 3 for the function defined by f(x) = x² - 3x + 5.

3. Show that the equation f(x) = (x³ + 2)/(x + 4) = 0 has a solution for x between 0 and 2.

4. Answer TRUE OR FALSE. Give a very brief explanation (if true) or a counterexample (if false) to support your answer.
(a) A continuous function f: (0,1) → R must have an absolute maximum or an absolute minimum.
(b) If |f|, the absolute value of the function f, is continuous at x = c then f is continuous at x = c.

5. (a) State the Location of Roots Theorem.
(b) Describe in your own words the idea of the proof of the Location of Roots Theorem.
(c) State Bolzano's Intermediate Value Theorem.
(d) Prove Bolzano's Intermediate Value Theorem using the Location of Roots Theorem.

6. (a) Find the first four terms of the sequence (aₙ) where a, aₙ = 3/5ⁿ⁻¹.
(b) Find the general term of the sequence: -1/4, 1/16, -1/64, 1/256, ?

7. (a) Write down the definition of what it means for limₙ→∞ 6n / (n + 5) = 6
(b) Given ε = 1/1000, find the smallest natural number K(ε), as in the definition you stated in part (2) above.
(c) Prove limₙ→∞ 6n / (n + 5) = 6

8. (a) Approximate the square root of 98 by finding s₃, using [sₙ₊₁ = (1/2)(sₙ + a/sₙ)], with initial guess of s₁ = 9. Leave your answer in reduced fraction form.
(b) How can you be sure that, if continued, the method in 3 (a) above will lead to a convergent sequence?

9. (a) State the definition of a Cauchy sequence.
(b) State what it means for a sequence not to be Cauchy.
(c) Prove that the sequence aₙ = (-1)ⁿ 1/5 is not Cauchy.

10. State whether the following is True or False.
Give a one sentence explanation, an example, or a counterexample to justify each answer.
(a) If a sequence is not bounded it is not Cauchy.
(b) A bounded sequence that is not monotone does not converge.
(c) The sequence defined by aₙ = (n⁵ + 7201n³ + 420n² + 10n + 5) / (n⁵ + 7201n³ + 400 n² + 10n + 6) has a convergent subsequence.

Unless otherwise stated, all functions are assumed to map the real...