Question

$\underset{x\to a}{lim}(x-a)(\frac{1}{x-a}-\frac{1}{x^2-(a+b)x+ab})$=______

• A. $\frac{a-b-1}{a-b}$
• B. 0
• C. 1
• D. $\frac{a+b}{a-b}$

Answer 1

The correct option is A $\frac{a-b-1}{a-b}$

Given, ${lim}_{x\to a}(x-a)(\frac{1}{x-a}-\frac{1}{x^2-(a+b)x+ab})={lim}_{x\to a}(\frac{(x-a)}{x-a}-\frac{(x-a)}{x^2-(a+b)x+ab})$
$={lim}_{x\to a}(1-\frac{(x-a)}{(x-a)(x-b)})={lim}_{x\to a}(1-\frac{1}{(x-b)})$
Substituting $x=a$ we get
$=1-\frac{1}{(a-b)}=\frac{a-b-1}{a-b}$