Yet another limit question in this forum (No.2 :P)

[Ballox]

Homework Statement

f(x)= lim t-> x [csc(t)-csc(x)]/(t-x). Find the value of f'(PI/4)

Homework Equations

I can see that this equation somewhat resembles one of the first principle def'ns

lim z->x [f(z)-f(x)]/(z-x)

The Attempt at a Solution

Not really sure how to begin.
I converted the csc(t) and the csc(x) to 1/sin(t) and 1/sin(x) and did a common denominator there, but I'm not sure if that's the correct approach to solving this problem.

I'm open to any suggestions and thank you for your time
Ballox

 

[tiny-tim]

Hi Ballox!

(have a pi: π )

Hint: one of the standard trigonometric identities is sin(t) - sin(x) = 2cos((t+x)/2)sin((t-x)/2)

 

[Ballox]

I don't recognize that as a standard trigonometric identity :(

 

[Mentallic]

Since it's of the indeterminate form 0/0, applying L'Hospital's rule should help

 

[Ballox]

Since it's of the indeterminate form 0/0, applying L'Hospital's rule should help

Haven't learned L'hopital's rule yet.
I believe we're supposed to use other methods

 

[tiny-tim]

(just got up :zzz: …)

I don't recognize that as a standard trigonometric identity :(

Then you will next time!

You can check for yourself that it's correct

(and familiarise yourself with the similar ones in the PF Library on trigonometric identities )

 

[HallsofIvy]

Homework Statement

f(x)= lim t-> x [csc(t)-csc(x)]/(t-x). Find the value of f'(PI/4)

Homework Equations

I can see that this equation somewhat resembles one of the first principle def'ns

lim z->x [f(z)-f(x)]/(z-x)

Quite correct- that is the "first principle" definition of the derivative of csc(x). So your problem is really "evaluate the second derivative of csc(x) at $x= \pi/4$".

The Attempt at a Solution

Not really sure how to begin.
I converted the csc(t) and the csc(x) to 1/sin(t) and 1/sin(x) and did a common denominator there, but I'm not sure if that's the correct approach to solving this problem.

I'm open to any suggestions and thank you for your time
Ballox

 

[Ballox]

Hmmm. So I read through all your responses and would like to thank you all for your help.

I looked back at the question and this is my solution:

limt->x [csc(t)-csc(x)]/(t-x) = d/dx csc(x) (not sure if this is represented correctly, but I see some sort of relationship here)

=> f(x)= -csc(x)cot(x)
f'(x)= -csc(x)*-csc^2(x) + cot(x)(csc(x)cot(x))
f'(x)= csc^3(x) + cot^2(x)(csc(x))

f'(PI/4)= 1/(sin^3(PI/4)) + [1/(tan^2(PI/4)) * 1/(sin(PI/4))]
f'(PI/4)= 2SQRT2 + (1*SQRT2)
f'(PI/4)= 2SQRT2 + SQRT2
f'(PI/4) = 3SQRT 2

Would this be correct?

 

[tiny-tim]

f'(PI/4) = 3SQRT 2

Would this be correct?

Yes.