limit as x approaches 1 of (sin(pi*x))/(x-1)?

Question

step by step details please without using derivatives or lehospital's rule just "identites, algebra, and trig" thanks. btw the answer is "-pi"
I need the steps

Answer 1

Evaluate lim sin(pi x)/(x-1)
x>1

Clearly this is indeterminate ie of the from 0/0. You can apply L'Hospital rule

lim g/h = lim g'/h' where g' and h' are the first derivatives

We get pi Cos(pi x)/1. As x goes to 1 the limit becomes pi x -1 or -pi

Don't you just love differential calculus

Answer 2

well sin pi goes to (Cos X) as for trig formulas. well i think we have to sue derivatives. I mean you should it looks like calculus to me. divison so it's vu1-uv1/v^2. It's too complicated for me I have AP Calculus and were working on Chapter 2 with derivatives. Hope this helps for a start