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If f(x)=cot^(-1) ((3x-x^3)/(1-3x^2)) and...

If `f(x)=cot^(-1) ((3x-x^3)/(1-3x^2))` and `g(x)=cos^(-1)((1-x^2)/(1+x^2))` then `lim_(x->a) (f(x)-f(a))/(g(x)-g(a))`

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`tantheta=a`
`0lttanthetalt1/2`
`0ltthetaltpi/6`
`x=tantheta`
`f(x)=cot^(-1)(tan3theta)=pi/2-tan^(-1)tan3theta`
`=pi/2-3theta`
`g(x)=cos^(-1)cos2theta=2theta`
`lim_(x->a)(f(x)*f(a))/(g(x)*g(a))*(x-a)/(x-a)=(f'(a))/(g'(a))`
...
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