If
$a > 0$ :
Limit[(x^a Log[x])/(Log[1 + x^2] (-Sin[x]^2 + Sin[x^2])), x -> 0, Direction -> "FromAbove", Assumptions -> a > 0]
(* ConditionalExpression[0, a > 6] *)
and
$0<a\leq 6$ then:
Limit[(x^a Log[x])/(Log[1 + x^2] (-Sin[x]^2 + Sin[x^2])), x -> 0, Direction -> "FromAbove", Assumptions ->0 < a <= 6]
(* ConditionalExpression[-\[Infinity], a < 0] *)