19: As x goes to negative infinity, then the ratio x²/(x − 1) tends to negative inifity. This is multiplied by sin(πx), which keeps oscillating between 1 and −1. Thus the whole expression oscillates between plus and minus huge numbers going to infinity; in other words, at negative infinity we have a never ending oscillation whose amplitude keeps growing. Such a situation means that there is no limit there.

Proof of this is best done using the Heine theorem, consider sequences x_n = π + 2kπ and y_n = −π + 2kπ.

Hint on limit at 1 from the left
Hint on limit at 1 from the right
Hint on limit at ∞
Answer