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
xn = π + 2kπ
and
yn = −π + 2kπ.
Hint on limit at
1 from the left
Hint on limit at
1 from the right
Hint on limit at
∞
Answer