Question

$\underset{x\to 1}{\lim }\frac{x^2-2x+1}{|x^2-1|}$ is equal to:

• A. 0
• B. 1
• C. $-1$
• D. 2
• E. does not exist

Answer 1

Answer: (A)

$\underset{x\to 1}{\lim }\frac{x^2-2x+1}{|x^2-1|}$ $|x^2-1|=\{\begin{array}{cc}-(x^2-1),& if-1<x<1\\ x^2-1,& ifx>1\end{array}$ $\therefore$ $\underset{x\to 1^{+}}{\lim }\frac{x^2-2x+1}{x^2-1}=\underset{x\to 1}{\lim }\frac{2x-2}{2x}=0$ $\underset{x\to 1^{-}}{\lim }\frac{x^2-2x+1}{1-x^2}=\underset{x\to 1}{\lim }\frac{2x-2}{-2x}=0$ $\therefore$ $\underset{x\to 1}{\lim }\frac{x^2-2x+1}{|x^2-1|}=0$