Limit

**hercules** · June 18th 2008, 07:27 AM

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

**TheEmptySet** · June 18th 2008, 07:51 AM

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 $

**hercules** · June 18th 2008, 07:59 AM

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

**TheEmptySet** · June 18th 2008, 08:06 AM

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.

**Chris L T521** · June 18th 2008, 08:08 AM

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:

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

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