Question

find the limit.
lim(x→1) sin(x - 1)/(x² + x - 2)

Explanation:

Step1: Factor the denominator

Factor $x^{2}+x - 2=(x - 1)(x+2)$. So the limit becomes $\lim_{x
ightarrow1}\frac{\sin(x - 1)}{(x - 1)(x + 2)}$.

Step2: Use the limit - property $\lim_{u

ightarrow0}\frac{\sin u}{u}=1$
Let $u=x - 1$. As $x
ightarrow1$, then $u
ightarrow0$. The limit is $\lim_{u
ightarrow0}\frac{\sin u}{u(u + 3)}$. We can rewrite it as $\lim_{u
ightarrow0}\frac{\sin u}{u}\cdot\lim_{u
ightarrow0}\frac{1}{u + 3}$.

Step3: Evaluate the limits

We know that $\lim_{u
ightarrow0}\frac{\sin u}{u}=1$ and $\lim_{u
ightarrow0}\frac{1}{u + 3}=\frac{1}{0 + 3}=\frac{1}{3}$. So the original limit is $1\times\frac{1}{3}=\frac{1}{3}$.

Answer:

$\frac{1}{3}$