1. ## Limit

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

2. Originally Posted by hercules
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
$\lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\cdot \frac{\sin(5x)}{7x}\right)$

We need to fix up the 2nd by making the denominator 5x

$
\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
$

3. Originally Posted by TheEmptySet
$\lim_{x \to 0} \left( \frac{\sin(4x+\pi)}{3}\cdot \frac{\sin(5x)}{7x}\right)$

We need to fix up the 2nd by making the denominator 5x

$
\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
$
Sorry, i guess i didn't write the problem clearly

$\lim_{x \to 0} \left( \frac{\sin((4x+\pi)/3)}{}\cdot \frac{\sin(5x)}{7x}\right)$

hope this is clearer (4x+pi)/3 is inside the sine

4. ## Version 2.0 Hehe

Okay here we go... The concept is the same.

$\lim_{x \to 0} \sin\left( \frac{(4x+\pi)}{3} \right) \cdot \left(\frac{\sin(5x)}{7x}\right)$

We need to fix up the 2nd by making the denominator 5x

$
\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.

5. Originally Posted by hercules
Sorry, i guess i didn't write the problem clearly

$\lim_{x \to 0} \left( \frac{\sin((4x+\pi)/3)}{}\cdot \frac{\sin(5x)}{7x}\right)$

hope this is clearer (4x+pi)/3 is inside the sine
Then it would be:

$\lim_{x\to{0}} \left[\sin\bigg(\frac{4x+\pi}{3}\bigg)\cdot\frac{5}{7}\f rac{\sin(5x)}{5x}\right]=\frac{\sqrt{3}}{2}\cdot\frac{5}{7}=\color{red}\bo xed{\frac{5\sqrt{3}}{14}}$