Question

find $lim_{t
ightarrow1}\frac{t^{2}+t - 2}{t^{2}-1}$. select the correct choice below and, if necessary, fill in the ans

a. $lim_{t
ightarrow1}\frac{t^{2}+t - 2}{t^{2}-1}=$ (type an integer or a simplified fraction.)
b. the limit does not exist.

Explanation:

Step1: Factor the numerator and denominator

The numerator $t^{2}+t - 2=(t + 2)(t - 1)$ and the denominator $t^{2}-1=(t + 1)(t - 1)$. So the function becomes $\lim_{t
ightarrow1}\frac{(t + 2)(t - 1)}{(t + 1)(t - 1)}$.

Step2: Cancel out the common factor

Since $t
eq1$ when taking the limit, we can cancel out the $(t - 1)$ terms. The limit simplifies to $\lim_{t
ightarrow1}\frac{t + 2}{t + 1}$.

Step3: Substitute $t = 1$

Substitute $t=1$ into $\frac{t + 2}{t + 1}$, we get $\frac{1+2}{1 + 1}=\frac{3}{2}$.

Answer:

A. $\lim_{t
ightarrow1}\frac{t^{2}+t - 2}{t^{2}-1}=\frac{3}{2}$