$\underset{x\to 1}{\lim }\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right)$ is equal

For $x\ne 1,f\left(x\right)=\frac{1}{1-x}-\frac{3}{1-x^3}$

$=\frac{1}{(1-x)}\left[1-\frac{3}{1+x+x^2}\right]$

$=\frac{1}{(1-x)}·\frac{x^2+x-2}{\left(1+x+x^2\right)}=\frac{(x+2)(x-1)}{(1-x)\left(1+x+x^2\right)}$

$=-\frac{x+2}{1+x+x^2}$

$\text{ so } \underset{x\to 1}{\lim }f\left(x\right)=-\underset{x\to 1}{\lim }\frac{x+2}{1+x+x^2}=-1$