Where it says Click for List you can skip this part.

(a) Julie is a 5-year old mathematical genius who has never attended any formal mathematics lectures. Recently, Julie discovered and proved the following theorem, naming it Julie's rule. Let k and h be two functions which are differentiable in a neighbourhood of some be R and lim k(x) = lim, h(x) = 0. Then, the existence of lim K (z) x-+bh'(x) implies that lim k(2) lim k (z) x-bh(x) x->bh'(x) However, Julie is still unsure about what may be concluded if lim K (z) x-bh'(x) does not exist. Julie is tempted to believe that the latter necessarily implies that lim k(z) x-+bh(x) does not exist. What would you tell Julie as a MATH1141 student? Answer: I would tell Julie that the implication is Click for List (b) Julie has a suspicion about what the answer to part (a) should be. In order to test the suspicion, Julie is trying to prove the following conjecture. Julie's conjecture. Let k, h and f be three functions which are differentiable in a neighbourhood of some b E R and lim, k(x) = lim h(x) = lim f(x) = 0. If lim k (z) c-+b f' (z) exists and lim f(2) x-bh(x) does not exist then nk(z) lim x-b h(x) does not exist. Do you think that Julie's conjecture is true? Answer: Click for List If you think that the conjecture is true then prove it in the essay belox below. If you think that it is wrong then provide a counterexample. uoneHB R defined by W for :1: 79 0 :12 c form=0, For) = where p is a fixed positive integer, n E Z and c is a real constant. (it Julie has found an integer N which is such that F is not continuous at :r: = 0 for all n > N regardless of the choice of c and continuous at a: = 0 for all n