Let $f(x) = \sin(x)$.  

1. Calculate the average rate of change from $x=\frac{\pi}{4}$ to $x=\frac{\pi}{2}$.

   Step (a):

   $\frac{s\left(\frac{\pi}{2}\right)-s\left(\frac{\pi}{4}\right)}{\left(\frac{\pi}{2}\right)-\left(\frac{\pi}{4}\right)}$

2. Write an expression for the average rate of change from $x=\frac{\pi}{4}$ to $x=\frac{\pi}{4}+h$.

   Step (b):

   $\frac{s\left(\frac{\pi}{4}+h\right)-s\left(\frac{\pi}{4}\right)}{\left(\frac{\pi}{4}+h\right)-\left(\frac{\pi}{4}\right)}$

3. Use the fact that $\sin\left(A + B\right) = \sin\left(A\right)\cos\left(B\right) + \cos\left(A\right)\sin\left(B\right)$ to show that the expression you wrote in part (b) can be rewritten as $\frac{1}{\sqrt{2}}\left(\frac{\cos(h)+\sin(h)-1}{h}\right)$.

   Step (c):

   $m=\frac{\text{sin}\left ( \frac{\pi }{4}+h \right )-\text{sin}\left ( \frac{\pi }{4} \right )}{h}=\frac{\text{sin}\left ( \frac{\pi }{4}\right )\text{cos}(h)+\cos\left ( \frac{\pi }{4} \right )\text{sin}(h)-\text{sin}\left ( \frac{\pi }{4} \right )}{h}$

   Substitute values for the trigonometric expressions and simplify.

4. Use the fact that $\lim\limits_{ x \rightarrow 0 } \frac { \operatorname { sin } ( x ) } { x } = 1$ and $\lim\limits_{ x \rightarrow 0 } \frac { \operatorname { cos } ( x ) - 1 } { x } = 0$ to evaluate $\lim\limits_{ h \rightarrow 0 } \frac { 1 } { \sqrt { 2 } } ( \frac { \operatorname { cos } ( h ) + \operatorname { sin } ( h ) - 1 } { h } )$.