Explain why the limit lim x→−∞ sin^−1 ((3 − x)/( x − 1)) is not well-defined.

Answer 1

lim x-> -∞ (3-x)/x-1) = -1

But for all values of x<0, (3-x)/(x-1)<-1 for which arcsin is not defined!

Answer 2

(3 - x)/(x - 1) < - 1; (3 - x)/(x - 1) + 1 < 0; 2/(x - 1) < 0; x - 1< 0; x < 1;

So for - ∞ < x < 1 sin^-1((3 - x)/(x - 1)) undefined