Question

Limit,
http://screencast.com/t/qDAYi1grkEw

Answer 1

oh its so much easy .. they have just differentiated denominator.

Answer 2

its L'hospital's rule

Answer 3

are you having problem in derivatives?

Answer 4

You want to find the derivative of \(\cot\left(\dfrac{1}{x}\right)\). Using the chain rule, you have
\[\frac{d}{du}\cot u\cdot\frac{d}{dx}\left(\frac{x}{2}\right)\]
where \(u=\dfrac{x}{2}\).
\[\frac{d}{du}\cot u=-\csc^2u=-\csc^2\left(\frac{x}{2}\right),~~\text{and }\frac{d}{dx}\left(\frac{x}{2}\right)=\frac{1}{2}\]
So,
\[\frac{d}{du}\cot u\cdot\frac{d}{dx}\left(\frac{x}{2}\right)=-\frac{1}{2}\csc^2\left(\frac{x}{2}\right)\]

Answer 5

nice explained bro :)

Answer 6

No I mean how did they find they end result: "2" I know how to find derivatives I am cofnused about the last step/result

Answer 7

As \(x\to\pi^+\), \(\csc \dfrac{x}{2}=\dfrac{1}{\sin\frac{x}{2}}\to\dfrac{1}{\sin\frac{\pi}{2}}=1\)

Answer 8

Then you're just left with \(\dfrac{1}{-\frac{1}{2}}=-2\).