# Limit problem with sin(x) /

1. Dec 10, 2012

### jkristia

limit problem with sin(x) / ....

1. The problem statement, all variables and given/known data

I'm stuck trying to algebraically find the limit of this expression shown below.
If I use L'Hospital I get the answer (√3) / 2 which is the correct answer, but it seems like no matter what I try I end up with 0/0.

Any hints of how to attack this is appreciated.

2. Relevant equations

3. The attempt at a solution

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2. Dec 10, 2012

### Dick

Re: limit problem with sin(x) / ....

I'm not sure why you'd want to approach it another way. Aren't you allowed to use l'Hopital? If not try substituting x=u+pi/6 and let u->0.

3. Dec 10, 2012

### jkristia

Re: limit problem with sin(x) / ....

Thank you for your suggestion, but even with substitution I keep going in circles. I will continue looking at this tomorow.

As for not using L'Hopital, I'm doing self study at the moment, working through problems in the book, and L'Hopital is not covered until another 7 chapters, so I figured I have to try and solve this without.

4. Dec 10, 2012

### Dick

Re: limit problem with sin(x) / ....

Use the trig rule sin(a+b)=sin(a)cos(b)+cos(a)sin(b). And you know the limits of sin(u)/u and (1-cos(u))/u, yes?

Last edited: Dec 10, 2012
5. Dec 10, 2012

### SammyS

Staff Emeritus
Re: limit problem with sin(x) / ....

Another possibility is to write 1/2 as sin(π/6) and then use a sum to product identity from trig. It still requires you to know the limit of sin(x)/x as x → 0 .

$\displaystyle \sin \theta - \sin \varphi = 2 \sin\left( \frac{\theta - \varphi}{2} \right) \cos\left( \frac{\theta + \varphi}{2} \right)$

6. Dec 11, 2012

### jkristia

Re: limit problem with sin(x) / ....

Thank you. I think replacing 1/2 with sin(pi/6) is what I was missing. I know the limit of sin(x)/x = 1.