Question

The function $f\left(x\right)=\frac{|x-3|}{x-3}$ at $x=3$, is

• A. Left continuous
• B. Right continuous
• C. continuous
• D. discontinuous

Answer 1

The correct option is D discontinuous

we just concentrate on the function $\frac{|x|}{x}$
LHL = lim $x\to 0^{-}\frac{|x|}{x}=\frac{-x}{x}=-1$

RHL = lim $x\to 0^{+}\frac{|x|}{x}=\frac{x}{x}=1$

Let , $y=x-3$
Now, if $x\to 3,then,y\to 0$
Applying the above derived result for y,
Since, the LHL is not equal to the RHL,
Hence the function is discontinuous at y = 0 or x = 3