Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
test the differentiability of the function at x=0 F(x)= {x*sin(1/x)if x≠0 and 0 if x=0
 2 years ago
 2 years ago
test the differentiability of the function at x=0 F(x)= {x*sin(1/x)if x≠0 and 0 if x=0
 2 years ago
 2 years ago

This Question is Closed

shankvee\Best ResponseYou've already chosen the best response.0
It is not differentiable. you can check that the function is continous. Applyinf frist principle \[f \prime(0)=\lim_{h \rightarrow 0}(f(h)f(0))/h=hsin(1/h)/h=\sin1/h\] sin1/h is undefined as h tends to 0 hence it is not differentiable.
 2 years ago

suju101Best ResponseYou've already chosen the best response.0
but can't we say 1/h>infinity
 2 years ago

shankvee\Best ResponseYou've already chosen the best response.0
Yeah since 1/h>infinty the limit sin(1/h) and subsequently f'(0) is undefined.
 2 years ago

shankvee\Best ResponseYou've already chosen the best response.0
If you have x^a sin(1/x) it is differentiable at x=0 only for a>1 for an integer a am not sure for fractional a value.
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
The derivative exists at x=0 if the limit of the difference quotient \[ \frac{f(h)  f(0)}{h} = \sin(1/h) \] as h > 0 exists. But it doesn't exist because \[ \lim_{h \rightarrow 0} \sin(1/h) \] doesn't exist. Instead, as h approaches 0, from either above or below, sin(1/h) oscillates faster and faster between 1 and 1. If it had a limit, L say, then we could make sin(1/h) as close as we liked to L by making h sufficiently close to zero. But on the contrary, for any restriction of h close to zero, sin(1/h) assumes infinitely many values between 1 and 1.
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.