Solving Two Limit Problems: Find \lim_{x\rightarrow 0, \frac{\pi}{4}}

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

Help!

 

[courtrigrad]

For (1) Use l'hospital's RuleFor (2) rewrite $$1-\tan x$$ as $$1 - \frac{\sin x}{\cos x}$$

 

[Checkfate]

I don't know l'hospital's 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.

 

[Checkfate]

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

 

[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...

 

[Checkfate]

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

 

[Checkfate]

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

 

[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.

 

[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...