Two limit problems

1. Oct 13, 2006

Checkfate

Hello,

I am having difficulties with two limit problems and hopefully someone can show me a trick or two that I don't know!

1) find $$\lim_{x\rightarrow 0}\frac{sin^{2}(\frac{x}{2})}{sin(x)}$$

Is it just me or is this thing a beast of a limit. I just can't get the denominator to equal anything other then 0!

I tried using the half angle identity on the top but it simply doesn't help...

2)find $$\lim_{x\rightarrow \frac{\pi}{4}}\frac{1-tan(x)}{sin(x)-cos(x)}$$ This one I don't even know where to start, I know that I need to get 1-cos(x)/x or cos(x)-1(x) and/or sin(x)/x or x/sin(x)... but I don't see a way.

PS- thats x approaches x/2(not pi....) sigh (for q1) and PI/4 (for q2) btw. it's small. :(

Help!

Last edited: Oct 13, 2006
2. Oct 13, 2006

For (1) Use L'Hopitals Rule

For (2) rewrite $$1-\tan x$$ as $$1 - \frac{\sin x}{\cos x}$$

3. Oct 13, 2006

Checkfate

I don't know L'Hopitals Rule. Is it easy to learn? If so I will just look it up and try to learn it :)

For q2, I will try it, thanks.

4. Oct 13, 2006

Checkfate

Learned it already, :) well, sorta, :) seems fairly straight forward.

5. Oct 13, 2006

quasar987

I'd be interested in knowing how can 1) be done without l'Hospital. The most natural first step would be to try the substitution

$$\sin^2(\frac{x}{2})=\frac{1-\cos x}{2}$$

but I don't see where to go from there...

6. Oct 13, 2006

Checkfate

Yea, it's beastly. I'll ask my teacher and post it :)

7. Oct 13, 2006

Checkfate

I used l'hospital's for both #1 and #2, :). Very straightforward and immensely useful!

8. Oct 13, 2006

quasar987

mmmh yea but I hope you're aware that if you haven't learned l'Hospital's rule in class yet, and use it in the exam, that's 0 for you. So you'd better learn to do the limits without it.

9. Oct 14, 2006

Checkfate

In the supplementary note package that came with my course it says that l'hospitals is outside the scope of the course, but says that it can be used. I just never bothered to learn it until now. But I do agree with you 110% that I need to do them both ways if I want to use l'hospitals. Not because I will necessarily lose marks, but because the practice is needed.
I think I will do a bunch of limit problems since I still can't do #1 without L'Hospitals...

Last edited: Oct 14, 2006