Discussion Forum



1)

Let f be the function defined by

f(x) = $\begin{cases}\frac{x^{2}-1}{x^{2}-2|x-1|-1} & x \neq 1\\1/2 & x = 1\end{cases}$


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:

Option D

Explanation:

For x< 1, f(x) = $\frac{x^{2}-1}{x^{2}+2x-3}$

= $\frac{x+1}{x+3}$

.'. $\lim_{x \rightarrow 1^{-}}$f(x) = 1/2

For x>1, f(x) = $\frac{x^{2}-1}{x^{2}-2x+1} =\frac{x+1}{x-1}$

.'. $\lim_{x \rightarrow 1^{+}}$ f(x) = ∞ 

.'. the function is not continuous at x= 1