Limit of a function

Question

Evaluate the following limits?

1) Lim t^2 + t - 2/ t^2 – 1                                  t→ 1
2) Lim -2x – 4/ x^3 + 2x^2                               t → - 2
3) Lim u^4 – 1 / u^3 – 1                                    u → 1
4) Lim √(x-3) / x – 9                                           x → 9
5) Lim x – 1/ √(x+3-2)                                        x → 1

Answer 1

(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.