Question

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

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