Question

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

Answer 1

Consider $\frac{1}{1-x}-\frac{3}{1-x^3}$

$=\frac{(1-x^3)-3(1-x)}{(1-x)(1-x^3)}$

$=\frac{(1-x)(1+x^2+x)-3(1-x)}{(1-x)(1-x^3)}$

$=\frac{(1-x)(1+x^2+x-3)}{(1-x)(1-x^3)}$

$=\frac{(x^2+x-2)}{(1-x^3)}$

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

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

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

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

Now ${lim}_{x\to 1}(\frac{1}{1-x}-\frac{3}{1-x^3})$

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

$=\frac{-(1+2)}{(1+1^2+1)}$

$=\frac{-3}{3}=-1$