I'm having my dumb hour.
lim (sin((4x+pi)/3)*sin(5x))/7x as x approaches zero
Simplify and show all work
we can use lim sin x/x -->1
what is the final answer supposed to be?
Thanks
$\displaystyle \lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\cdot \frac{\sin(5x)}{7x}\right) $
We need to fix up the 2nd by making the denominator 5x
$\displaystyle
\lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\right)\cdot \left(\lim_{x \to 0} \frac{5}{7} \left[ \frac{\sin(5x)}{5x}\right]\right)=(0)\cdot \left( \frac{5}{7}\left[1\right] \right)=0
$
Okay here we go... The concept is the same.
$\displaystyle \lim_{x \to 0} \sin\left( \frac{(4x+\pi)}{3} \right) \cdot \left(\frac{\sin(5x)}{7x}\right) $
We need to fix up the 2nd by making the denominator 5x
$\displaystyle
\lim_{x \to 0}\sin \left( \frac{(4x+\pi)}{3}\right)\cdot \left(\lim_{x \to 0} \frac{5}{7} \left[ \frac{\sin(5x)}{5x}\right]\right)=(\sin\left( \frac{\pi}{3}\right))\cdot \left( \frac{5}{7}\left[1\right] \right)=\left( \frac{\sqrt{3}}{2}\right)\cdot \left( \frac{5}{7}\right)=\frac{5\sqrt{3}}{14}
$
Good luck.