=  +  =  + .    (1)


Therefore,  sin((pi*x)/(x-1)) = sin(pi + pi/(x-1)) = .    (2)


When x→ + ∞,  (x-1) is large value;   is a small value;

therefore,  is a small value equivalent to . 


It implies that    is a negative value equivalent to  .


Then    is equivalent to  .


As x→ + ∞,  the quantity    tends to -2.


Thus limit   is  -2  when  (x→ + ∞).