limits problem

limits problem

Postby Guest » Mon Jan 25, 2021 4:38 am

find the limit of this function [tex]\lim_{x \to а}(\frac{tg x}{tg a})^{cotg(x-a)}[/tex] , [tex]a\ne \frac{k\pi}{2}[/tex] , [tex]k\epsilon Z[/tex]

Re: limits problem

Postby HallsofIvy » Fri Jan 29, 2021 5:08 pm

Take the logarithm: [tex]log\left(\frac{tan(x)}{tan(a)}\right)^{cot(x- a)}= cot(x- a)log\left(\frac{tan(x)}{tan(a)}\right)= cot(x- a)log(tan(x)- cot(x- a)log(tan(a)[/tex]

I think you should reduce that cot(x- a): [tex]cot(x- a)= \frac{cot(x)cot(a)+ 1}{cot(x)- cot(a)}[/tex].

So this beco{mes [tex]\frac{(cot(x)cot(a)+ 1)log(tan(x))}{cot(x)- cot(a)}[/tex][tex]- \frac{(cot(x)cot(a)+1)log(tan(a))}{cot(x)- cot(a)}[/tex]

Differentiating that is tedious but elementary in every step- use the quotient rule, the chain rule, the derivatives of tangent and cotangent, and logarithm.

Once you have found the derivative of this function the derivative of [tex]\left(\frac{tan(x)}{tan(a)}\right)^{cot(x- a)}[/tex] is the exponential of that.

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