Questions 1. * Let x and y be positive real numbers with x < y. Prove that x2 < y2. 2. * Prove that if a < b and c < d then a + c < b + d. 3. * 1 Prove that the sequence tends to zero. n 4. * (a) Prove that if 1 = 0 n n a n (b) Give a counterexample to show that the converse (if lim a n = 0 then lim a n = then lim n 1 = ) is false. lim n a n 5. * Give an example of a sequence {an} satisfying all of the following: {an} is monotonic 0 < an < 1 for all n and no two terms are equal 1 lim a n = 2 n 6. * Let k > 0 be a constant and consider the important sequence {kn}. It’s behaviour as n will depend on the value of k. (i) State the behaviour of the sequence as n when k = 1 and when k = 0. (ii) Prove that if k > 1 then kn as n (hint: let k = 1 + t where t > 0 and use the fact that (1 + t)n > 1 + nt. (iii) Prove that if 0 < k < 1 then kn 0 as n . 7. * Given a geometric series with first term a > 0 and common ration r > 0 prove that a finite “sum to infinity” exists if and only if r < 1 and show that in this case the a sum to infinity is . 1 r (hint: use the results of Q6 above, and recall that the sum of the first n a (1 r n ) terms is ). 1 r 8. * Use the definition of f(x) as x to show that f(x) = x + 2 as x . 9. * (a) Show that f(x) = 3 tends to as x –2+ . x2 1 (b) Show that f(x) = 1 tends to as x 4– 4x 4 when x 5 10. * Let f(x) = . 0 when x 5 Show that f is not continuous at x = 5. 11. * Prove that if f is continuous at all values of x then so is kf where k is a constant. 12. * Give an example of a function that is continuous at all non-integer values but is discontinuous at all integer values. Prove the discontinuity property of your function (i.e. that it is discontinuous at all integer values). 13. * Explain why this is no good as a definition of continuity at a point a (either by giving an example of a continuous function that does not satisfy the definition or a discontinuous one that does): Given > 0 there exists a > 0 such that |x – a| < |f(x) – f(a)| < 14. * Can a function be continuous at one value of x and discontinuous at all other x R? Explain your answer, giving proofs where appropriate. 15. * Show f(x) = 2 is not uniformly continuous on the interval (0, 1). x . 16. * Prove that if f and g are functions that are uniformly continuous on an interval I then so is f + g. (Hint: a proof very similar to that for Example 3 will work, you just have to adapt it to the definition of uniform continuity). 1 1 17. * Show that if f(x) = then f (c) = . 3 x 2 c 1 x 2 18. * Show that the function f(x) = 1 x when x 0 is not differentiable at x = 0. when x 0 19. * What can you say about the differentiability of fg at x = c in each of the following cases? (Here the function fg is defined by fg(x) = f(x)g(x)). (a) f is differentiable at c, but g is not. (b) f is not differentiable at c, and neither is g. (c) both f and g are differentiable at c. 2 20. * Let f(x) = 5 x . Show that for c 0, f is differentiable at c, with f (c) = 1 5 5 c4 , but that f is not differentiable at 0. (You will find the identity a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4) useful). 21. * Prove that between any two roots of x4 – 4x2 + k = 0 there is at least one root of x3 = 2x. 22. * Suppose f is continuous and differentiable on [0, 1], that 0 f(x)) 1 for each x in [0, 1], and that f (x) 1 each x in [0, 1]. Show that there is at most one solution of f(x) = x in [0, 1]. 23. * In the Cauchy Mean Value Theorem take f(x) = x2, g(x) = 4x3 – 3x4 on the interval [0, 1]. Show that the conditions of the theorem hold and find possible values of "c". 3