Question

\[\lim_{x \to \frac{\pi}{4}} \frac{f\left( x \right) - f\left( \frac{\pi}{4} \right)}{x - \frac{\pi}{4}},\]

Answer 1

\[\lim_{x \to \frac{\pi}{4}} \frac{f\left( x \right) - f\left( \frac{\pi}{4} \right)}{x - \frac{\pi}{4}}\]
\[ = \lim_{h \to 0} \frac{f\left( \frac{\pi}{4} + h \right) - f\left( \frac{\pi}{4} \right)}{\frac{\pi}{4} + h - \frac{\pi}{4}}\]
\[\text{ It is given that } f\left( x \right) = \sin 2x . \]
\[ \Rightarrow \lim_{h \to 0} \frac{\sin \left( \frac{\pi}{2} + 2h \right) - \sin \left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\cos 2h - 1}{h}\]
\[ = \lim_{h \to 0} - 2\left( \frac{\sin^2 h}{h \times h} \right) \left( h \right)\]
\[ \Rightarrow \lim_{h \to 0} \left( - 2h \right) = 0\]