(1) t2+t-2=(t+2)(t-1); t2-1=(t+1)(t-1); so when t≠1, (t2+t-2)/(t2-1)=(t+2)/(t+1). As t→1 this approaches 3/2, so 3/2 is the limit.
(2) (-2x-4)/(x3+2x2)=-2(x+2)/(x2(x+2))=-2/x2 when x≠-2 (question should read x, not t). As x→-2, this becomes -2/4=-½, which is the limit as x→-2.
(3) (u4-1)/(u3-1)=(u-1)(u3+u2+u+1)/[(u-1)(u2+u+1)]=(u3+u2+u+1)/(u2+u+1) when u≠1. This evaluates to 4/3, therefore as u→1, the limit is 4/3.
(4) √(x-3)/(x-9). When x=9+e (e>0), this becomes √(6+e)/e (large positive when e small), and when x=9-e, it becomes √(6-e)/(-e) (large negative when e small). So the left and right limits of the expression are not the same, and the limit cannot be defined.
(5) (x-1)/√(x+3-2)=0/2=0 when x=1, so the limit is zero. The limit is -∞ as x→-1.