# Thread: Csc Limit question

1. ## Csc Limit question

Find $\displaystyle f'(\frac{\pi}{4})$ if $\displaystyle f(x)=\lim_{t\to{x}}\frac{csc\;t -csc\;x}{t-x}$

2. Originally Posted by polymerase
Find $\displaystyle f'(\frac{\pi}{4})$ if $\displaystyle f(x)=\lim_{t\to{x}}\frac{csc\;t -csc\;x}{t-x}$
note that $\displaystyle f(x)$ is defined by the limit definition of the derivative of $\displaystyle \csc x$

thus, let $\displaystyle g(x) = \csc x$

we have that $\displaystyle f(x) = g'(x)$

so $\displaystyle f'(x) = g''(x)$

therefore, $\displaystyle f' \left( \frac {\pi}4 \right) = g'' \left( \frac {\pi}4 \right)$

now find $\displaystyle g''(x)$ and you will have your answer

are you required to find the derivative by evaluating the limit? i doubt it. it will especially be a pain doing it for the second derivative

3. Originally Posted by Jhevon
note that $\displaystyle f(x)$ is defined by the limit definition of the derivative of $\displaystyle \csc x$

thus, let $\displaystyle g(x) = \csc x$

we have that $\displaystyle f(x) = g'(x)$

so $\displaystyle f'(x) = g''(x)$

therefore, $\displaystyle f' \left( \frac {\pi}4 \right) = g'' \left( \frac {\pi}4 \right)$

now find $\displaystyle g''(x)$ and you will have your answer

are you required to find the derivative by evaluating the limit? i doubt it. it will especially be a pain doing it for the second derivative
maybe im just doing it wrong but i'm getting $\displaystyle \sqrt{2}+\sqrt{8}$ and thats not right