evaluate the following limit $$\lim_{x \rightarrow 0^{+}}\cot x-\frac{ 1 }{ x }$$

Question

anonymous · Answer

I know that it is infinity-infinity but I don't know how to change it into a problem I can work with

terenzreignz · Answer

Well... How about this...
$$\huge \cot \ x - \frac1{x}=\frac{x\cos \ x-\sin \ x}{x\sin \ x}$$Now it's just a fraction.
L'hopital away, champ ^.^

anonymous · Answer

wow, thanks

terenzreignz · Answer

o.O
Here I am again, overcomplicating things... maybe you could try this, it might be a lot less headache-inducing...
$$\huge \cot x-\frac1{x}=\frac{x\cot \ x -1}{x}$$

terenzreignz · Answer

Sorry about the first one, this one only requires one use of L'Hopital...

terenzreignz · Answer

Or then again, maybe not.  I don't know anymore O.o

anonymous · Answer

well when I did the first one i simplified it down to -sinx/cosx and substituted zero and got 0 which is the correct answer in the book.  But when you're second guessing yourself I want to make sure I did it correctly

terenzreignz · Answer

Never mind the second... I tried... it loses more than it gains... it gets a cot x in the numerator, which is troublesome.  Stick to my first idea :)
Instincts are to be trusted, after all :)

anonymous · Answer

lol alright i get it, happens to me all the time.

terenzreignz · Answer

Good hunting :)