Question

find the limit as x approaches 0 of (csc(pi/6+deltax)-cscpi/6)/deltax

Answer 1

Im lost on where to start because the denominator can not be 0

Answer 2

\[\lim_{\Delta x\to0}\frac{\csc\left(\dfrac{\pi}{6}+\Delta x\right)-\csc\dfrac{\pi}{6}}{\Delta x}\]
\[\lim_{\Delta x\to0}\frac{\sin\dfrac{\pi}{6}-\sin\left(\dfrac{\pi}{6}+\Delta x\right)}{\Delta x\sin\left(\dfrac{\pi}{6}+\Delta x\right)\sin\dfrac{\pi}{6}}\]
\[\lim_{\Delta x\to0}\frac{\dfrac{1}{2}-\sin\dfrac{\pi}{6}\cos\Delta x-\cos\dfrac{\pi}{6}\sin\Delta x}{\dfrac{1}{2}\Delta x\left(\sin\dfrac{\pi}{6}\cos\Delta +\cos\dfrac{\pi}{6}\sin\Delta x\right)}\]
\[\lim_{\Delta x\to0}\frac{\dfrac{1}{2}-\dfrac{1}{2}\cos\Delta x-\dfrac{\sqrt3}{2}\sin\Delta x}{\dfrac{1}{2}\Delta x\left(\dfrac{1}{2}\cos\Delta +\dfrac{\sqrt3}{2}\sin\Delta x\right)}\]
\[2\lim_{\Delta x\to0}\frac{1-\cos\Delta x-\sqrt3\sin\Delta x}{\Delta x\left(\cos\Delta +\sqrt3\sin\Delta x\right)}\]
\[2\left(\lim_{\Delta x\to0}\frac{1-\cos\Delta x}{\Delta x}-\sqrt3\lim_{\Delta x\to0}\frac{\sin\Delta x}{\Delta x}\right)\left(\lim_{\Delta x\to0}\frac{1}{\cos\Delta x+\sqrt3\sin\Delta x}\right)\]

Answer 3

thank you so much