Question

Determine lim sin(x)/(x²-x)

Answer 1

The limit of a product is equal to the product of limits:

$\displaystyle\lim_{x\to0}\frac{\sin x}{x^2-x}=\left(\lim_{x\to0}\frac{\sin x}x\right)\left(\lim_{x\to0}\frac1{x-1}\right)$

You should recognize the first limit, which has a value of 1. In the second limit, the expression $\dfrac1{x-1}$ is continuous at $x=0$, so its value is -1, and so the overall limit has a value of 1*(-1) = -1.