This problem contains a specal case of L'Hopitals rule (a) Let h: S→ R and c be a cluster point of S. Show that if lim h(x) = L = 0, then thereexists some 8 >0 such that for all x € (S\ {c}) n (c-8, c+8), h(x) ‡ 0.2-c(b) Let h: S→ R be continuous and c be a cluster point of S. Show that if h(c) # 0, thenthere exists some AC S such that c is a cluster point of A, h|₁ (x) ‡0 for all x ¤ A,and1lim (MAP))-I-C h|A(x)1lim (h(x)) h(c)2-c¹Most trigonometric identities and inequalities have "geometric" proofs, so it doesn't count as "cheat-ng" to use them to prove facts about calculus. See https://en.wikipedia.org/wiki/Proofs_of_rigonometric_identities for example.limI-CNote: This result allows us to "abuse notation". We get a slightly more general notionof Corollary 3.1.12.iv and write=(h(x))1lim h(x)I-Ceven though strictly speaking, 1/h(x) might not be defined for all x € S.

Answered: Image /qna-images/answer/dd026aa6-ae83-4ccd-afa7-454287c5c937.jpg

(a) Let h: S→ R and c be a cluster point of S. Show that if lim h(x) = L = 0, then there exists some > 0 such that for all x € (S\ {c}) n (c-8, c+d), h(x) ‡ 0. 2-c (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 AC S such that c is a cluster point of A, h|₁ (x) ‡0 for all x ¤ A, and 1 lim (ALA(2)) - 1 lim (h(x)) h(c) I-C =

(a) Let h: S→ R and c be a cluster point of S. Show that if lim h(x) = L = 0, then there exists some > 0 such that for all x € (S\ {c}) n (c-8, c+d), h(x) ‡ 0. 2-c (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 AC S such that c is a cluster point of A, h|₁ (x) ‡0 for all x ¤ A, and 1 lim (ALA(2)) - 1 lim (h(x)) h(c) I-C =

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.1: Limits

Problem 66E

See similar textbooks

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
What is the purpose of the Intermediate Value Theorem?
Evaluate the limit 8x2 – 9x – 9 - lim x→-o 7 – 7x – 10x2 If the limit does not exist enter DNE. Limit
If f : A → R, c E R is cluster point and lim|f(x)| = L, then „lim f(x) = L إختر واحداً:
7 - Let F(t) = cosnti³ + sinntj³ + 6tk¬ G(t)* = 2ti* + t³j> Find a) lim[F*.G¯](t) b) lim[F x G*l(t) t-3 t-3
2. Use the paths x = y and x = 0 to verify the limit. 4x²y = 0 (x,y)-(0,0) x2+y2, lim
Prove that lim r(x) = 0 where m = 0,1, 2, 3, ... X-m
Prove that lim x 2 = 4. x→ 2
b) The function g is defined by x-2 x+2 Show that lim_g(x) exist. x→ -2 4+
Find the limit V1+t4 dt lim
f(x) – f(c) If lim exists, it is called the of f at c.