3. Show that the following two definitions of differentiability are equivalent.(a) The function f is differentiable at r = a if and only if the limitL = limAr-0f(x+ Az) – f(x)Arexists. In this case we define f'(a) = L.(b) The function f is differentiable at z = a if and only if there is a constant m and a function E of z, defined forall z + a satisfying the two conditionsf(x) = f(a) + m(x = a) + E(x)(z – a) for all z # a,lim E(r) = 0.In this case we define f'(a) = m.

Answered: Image /qna-images/answer/0994a8cd-fdce-4bc7-b38d-6ab28a94d608.jpg

Answered: (n) The function f is differentiable at…

(n) The function f is differentiable at z = a if and only if the limit L = lim {(z+Az) – f(x) Ar Ar0 exists. In this case we define f'(a) = L. %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: . For an arbitrary function f, we define f*(=) = lim h→0 f(a+h)-f(z-h) 2h if this limit exists. Let…

Q: xy3 lim (x.y)-(0,0) x* + y+

A: Here we go towards origin via two different straight ling and got two different limit so the limit…

Q: The limit lim h→0 f(x) = ab a= b sin (a) sin (a) In(e +h)-1 h 8 x α a represents f'(a) for some…

A: Topic:- application of derivatives and calculus

Q: fn cte USE VOUR SMARTPHONE FOR Reviews ideos- Features Specs - SuPport SCAN Standard data rates may…

A: In part a we will differentiate using product rule of differentiation. Hindi movie in part be we…

Q: Which of these is the limit definition of the derivative of f(x)? O f'(x) f (x + h) – f (x) lim h 00…

A: use the basic definition of derivative to choose the correct option

Q: Consider the following functions. (a) f: R*→ GL2 (R), f(a) = 1) (b) ƒ: R* → GL₂ (R), f(a) = (i). a…

Q: 2. Let f(x) be a continuous function on R. The following is known about f(x): • The domain of f(x)…

Q: (b) With the help of Sandwich theorem (or Squeeze theorem), show that lim (2 √√x - 1)" = x² for x ≥…

A: By using Sandwich theorem, we have to show that ; limn→∞2xn-1n = x2 for x≥1

Q: ƒ(0) = 0, and ƒ'(−2) = f'(1) = ƒ'(9) = 0 ● lim f(x) = 0 and lim f(x) x →∞ x→6 • f'(x) 0 on (-2, 1)…

Q: Sketch the graph of a function that satisfies all of the following conditions. Explair • f(0) = 0,…

Q: f(z) = 7. Use f'(x) = lim e+h-©) to find f'(x). %3D %3D 3z-7

A: The given funciton is:fx=23x-7To find f'x

Q: Let f: RR be a function that is differentiable on (a, o), where a is any real constant. Let g: RR be…

Q: (3) Find the limit if it exists. 1' - ry? lim (a) (z.y)-(0,0) r2 + y? (b) lim (2 + y*) In (x² + y²)…

A: Now we will find the solution of the given limit.

Q: (4) Find the limit by interpretting the limit as as an appropriate derivative. (DO NOT USE…

Q: (5) Let f: R→R, and suppose there exist a positive integer M such that 0 < f{x) < M for all x ER.…

Q: Find f'(a) using the limit definition for 1 f(x) = V6 – 5x -

Q: If f and g are continuous functions with f(-1) = 3 and the following limit, find g(-1) _lim [2f (x)…

Q: (a) Let {n} be a sequence such that there exist A> 0 and c € (0, 1) for which n+1- n ≤ Ac" for all…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: 15. Suppose that the functionf:R Ris differentiable at xg. Prove that xf (xo)- xof (x) lim = f(xo) -…

Q: 1. Evaluate the following limits lim x(/x+1-Vx) 2. Find f'(x) of each of the following functions (x)…

A: As per our guidelines we need to solve only one questions and please repost it as other questions.

Q: Consider the function (a) Compute the derivative of f(x) directly from the limit definition.…

A: We will use the given information to solve the given questions

Q: Find the lim ex-y limit or (xy)-> (0,1) 2 cosx-1-y Show that it does not exist. DNES

A: To find the limit .To show that it does not exist.

Q: Let f(x) be a differentiable function with the properties that f(1) = 8 and lim f(x) = 5. [ªfa f'(x)…

A: f(x) be a differentiable function with the propertiesf(1)=8 and limx→∞f(x)=5.

Q: The limit lim h→0 f(x) = cos(h) - 1 h represents f' (a) for some function f and some number a. Find…

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

100%

3. Show that the following two definitions of differentiability are equivalent.
(a) The function f is differentiable at r = a if and only if the limit
L = lim
Ar-0
f(x+ Az) – f(x)
Ar
exists. In this case we define f'(a) = L.
(b) The function f is differentiable at z = a if and only if there is a constant m and a function E of z, defined for
all z + a satisfying the two conditions
f(x) = f(a) + m(x = a) + E(x)(z – a) for all z # a,
lim E(r) = 0.
In this case we define f'(a) = m.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Limits and Continuity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

true or false
1. Let f: (0,00)→ R be a differentiable function such that lim f'(x) = 0. x+00 Prove or disprove that lim [f(x + 1)-f(x)] = 0.
f (a + h) – f(x) Find f'(3) using f'(a) = lim for the following function h→0 f(x) = 2x2 + 3x – 5 %3D
Given a function f(x) = c, where c is some constant. Show that f'(x) = 0. (Hint: by limit definition.)
Sketch the graph of the function f that has this properties
Find the limit: (x +1)²(x–1) -1) lim メ→-1 .3 x'+1
Suppose lim h→0 TRUE? ƒ(1 + h) − ƒ(1) h = -4. Which are the following statements are f'(1)=-4 f(x) is differentiable at x=1 f(x) is continuous at x=1 f'(0)=-4 f(x) is differentiable for all values of x f(1)=-4
4. Let f(x) = x² sin() and g(x) = sinx. Prove that f(x) lim z>0 g(x) Is it true that f(x) lim T >0 g(x) exists. f'(x) lim I >0 g'(x) ?
Sketch the graph of a function that satisfies the given conditions. r loirtw mo alern doidw ao slar atiog noiloellari f(0) = 0, f'(-2) = f'(1) = f'(9) = 0, lim f(z) = 0, lim f(x) =-00, f'(z) 0 on (-2, 1) and (6,9), f"(z) > 0 on (-∞,0) and (12, o0), f"(z) <0 on (0, 6) and (6, 12).
The following table includes partial information for the values of a differentiable function of and its derivatives. 3 5 f(x) 0 20 f'(x) 5 2 f"(x) 7 What is the value of the following limit? O O ƒ(f'(2x - 3)) lim x3 3x² 10x + 3 - 74 72 35 8 This information is insufficient to determine the limit.
Please provide a clear and complete solution. Answer fast for I have 30 minutes left. Thank you very much.
True or False Suppose that f'(x) is a function that is differentiable everywhere except (a) for x = 1. Then f(1+h) – f(1) lim h→0 h exists. Note: We're not trying to compute this. Just determining whether or not this exists. (b) If f(x) is continuous everywhere, then f(x) is differentiable everywhere.