Question

question evaluate the limit and write your answer in simplest form: $lim_{h \to 0}\frac{csc(\frac{pi}{3}+h)-csc(\frac{pi}{3})}{h}$

Explanation:

Step1: Recall the definition of the derivative

The given limit $\lim_{h
ightarrow0}\frac{\csc(\frac{\pi}{3}+h)-\csc(\frac{\pi}{3})}{h}$ is in the form of the derivative of the function $y = \csc(x)$ at $x=\frac{\pi}{3}$. The derivative of $y=\csc(x)$ is $y'=-\csc(x)\cot(x)$.

Step2: Substitute $x = \frac{\pi}{3}$ into the derivative formula

We know that $\csc(\frac{\pi}{3})=\frac{2}{\sqrt{3}}$ and $\cot(\frac{\pi}{3})=\frac{1}{\sqrt{3}}$.
Substitute into $y'=-\csc(x)\cot(x)$, we get $y'|_{x = \frac{\pi}{3}}=-\frac{2}{\sqrt{3}}\times\frac{1}{\sqrt{3}}$.

Step3: Simplify the result

$-\frac{2}{\sqrt{3}}\times\frac{1}{\sqrt{3}}=-\frac{2}{3}$.

Answer:

$-\frac{2}{3}$