Question

Evaluate the limit:
$\underset{x\to 1}{lim}(\frac{1}{x-1}-\frac{2}{x^2-1})$

Answer 1

Given: $\underset{x\to 1}{lim}(\frac{1}{x-1}-\frac{2}{x^2-1})$

$becomes(\infty -\infty )form$

Using Factorization method

$\Rightarrow \underset{x\to 1}{lim}(\frac{1}{x-1}-\frac{3}{x^2-1})$

$[\because (a^2-b^2)=(a+b\left)\right(a-b\left)\right]$

$\Rightarrow \underset{x\to 1}{lim}(\frac{1}{x-1}-\frac{2}{(x+1)(x-1)})$

$\Rightarrow \underset{x\to 1}{lim}\left(\frac{(x+1)-2}{(x+1)(x-1)}\right)$

$\Rightarrow \underset{x\to 1}{lim}\left(\frac{x-1}{(x+1)(x-1)}\right)$

$\Rightarrow \underset{x\to 1}{lim}\left(\frac{1}{x+1}\right)$

$=\frac{1}{1+1}=\frac{1}{2}$

$Therefore,$

$\Rightarrow \underset{x\to 1}{lim}(\frac{1}{x-1}-\frac{3}{x^2-1})=\frac{1}{2}$