Problem Finding the limit with 2 parameters where $I(n,m) = \lim\limits_{x \to \pi} \frac{\sin (nx)}{\sin (mx)}$

Question

Here i have a problem since x approaches pi

So here i have to calculate the Limit and then add it

I (2019,2020) + I (2018,2020) + I (2019,2021) rounded upto 4 decimal places

I got a answer of 2.9975 which is wrong i need to know how to approach this problem

Answer 1

Hospital:

$\lim_{x \rightarrow π}\dfrac{n\cos (nx)}{m\cos(mx)}=$

$(n/m)(-1)^n(-1)^m=$

$(n/m)(-1)^{n+m}.$

Answer 2

Let $x-\pi=y$

$$\sin(mx)=\sin m(\pi+y)=\sin(my)\cos(m\pi)$$

Now for integer $m,\cos(m\pi)=(-1)^m$

For integer $m,n;$ $$\lim_{x\to\pi}\dfrac{\sin mx}{\sin nx}=(-1)^{m-n}\dfrac mn\cdot\dfrac{\lim_{y\to0}\dfrac{\sin my}{my}}{\lim_{y\to0}\dfrac{\sin ny}{ny}}$$