Question

Problem 5 (4 points each) In this problem, we will prove a special case of L'Hôpital's rule. a = cc (a) Let h: S → R and c be a cluster point of S. Show that if lim h(r) = L + 0, then there exists some d > 0 such that for all x € (S\{c}) n(c-,c+0), h(x) +0. (b) Let h: S → R be continuous and c be a cluster point of S. Show that if h(c) + 0, then there exists some A CS such that c is a cluster point of A, hlA (x) + ( for all x € A, and 1 lim c lim (h|4(x)) h(c) 1 () (late) bl() ne A(x) c Note: This result allows us to "abuse notation". We get a slightly more general notion of Corollary 3.1.12.iv and write 1 lim x+c (h(x) (0) - lim h(x) 1-> : even though strictly speaking, 1/h(x) might not be defined for all x E S. (C) Suppose f : (a, b) → R and g : (a, b) → R are differentiable functions whose derivatives + f' and g' are continuous functions. Suppose that at ce (a,b), f(c) = g(e) = 0, and g'(x) + 0 for all x € (a,b), and suppose that the limit of f'.) g' (2) as x + c exists. Show that f(x) f'(x) lim lim g(x) g'(x) 10 2-c (Hint: This is similar to the proof that a differentiable function is continuous. Be careful not to divide by 0, and make sure to explain all the steps in your proof.)

Answer 1

The question pertains to proving a special case of L'Hôpital's rule where one must demonstrate the behavior of functions and their limits around cluster points or points where they vanish.

Step-by-step explanation:

Special Case of L'Hôpital's Rule

The question is asking to prove a special case of L'Hôpital's rule in different scenarios:

For a function h, given that lim h(r) = L + 0 exists for r approaching some cluster point c, we must show that there is a delta (d) such that for all x in a punctured neighborhood around c, h(x) + 0.

For h continuous at a cluster point c and with h(c) + 0, one must find a subset A with c as a cluster point and show that 1/h has a limit as x approaches c within A.

For differentiable functions f and g with f(c) = g(c) = 0 and non-zero g'(x), and if the limit of f'(x)/g'(x) as x approaches c exists, then the limit of f(x)/g(x) as x approaches c also exists and is equal to the limit of f'(x)/g'(x).

The function h(x) is considered in different contexts to discuss the implications of limits and continuity at a point. For the differentiable functions f and g, their behavior around a point where they both vanish is investigated, utilizing their derivatives' limits to infer the limit of their ratios.