Question

Let $f$ be the function defined by
$f(x)=\{\begin{array}{ll}\frac{x^2-1}{x^2-2|x-1|-1},& x\ne 1\\ \frac{1}{2},& x=1\end{array}$

• A. The function is continuous for all values of $x$
• B. The function is continuous only for $x>1$
• C. The function is continuous at $x=1$
• D. The function is not continuous at $x=1$

Answer 1

Answer: (D)

For $x<1,f(x)=\frac{x^2-1}{x^2+2x-3}=\frac{x+1}{x+3}$
$\therefore \underset{x\to 1^{-}}{\lim }f(x)=\frac{1}{2}$
For $x>1,f(x)=\frac{x^2-1}{x^2-2x+1}=\frac{x+1}{x-1}$
$\therefore \underset{x\to 1^{-}}{\lim }f(x)=\infty$
$\therefore$ The function is not continuous at $x=1$