# limits problem

### limits problem

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

### Re: limits problem

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

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

So this beco{mes $$\frac{(cot(x)cot(a)+ 1)log(tan(x))}{cot(x)- cot(a)}$$$$- \frac{(cot(x)cot(a)+1)log(tan(a))}{cot(x)- cot(a)}$$

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 $$\left(\frac{tan(x)}{tan(a)}\right)^{cot(x- a)}$$ is the exponential of that.

HallsofIvy

Posts: 341
Joined: Sat Mar 02, 2019 9:45 am
Reputation: 123

### Who is online

Users browsing this forum: No registered users and 1 guest