Limits Continuity And Differentiability Question 114
Question: Which one of the following is correct in respect of the function $ f(x)=\frac{x^{2}}{| x |} $ for $ x\ne 0 $ and f(0) = 0?
Options:
A) f (x) is discontinuous every where
B) f (x) is continuous every where
C) f(x) is continuous at x = 0 only
D) f(x) is discontinuous at x = 0 only
Show Answer
Answer:
Correct Answer: B
Solution:
$ f(x)= \begin{matrix} \frac{x^{2}}{x}, & x\ne 0 \\ 0 & x=0 \\ \end{matrix} . $
$ = \begin{matrix} \frac{x^{2}}{x}=x, & x>0 \\ 0, & x=0 \\ \frac{x^{2}}{-x} =-x, & x<0 \\ \end{matrix} . $ Now, $ \underset{x\to {0^{-}}}{\mathop{\lim }}f(x)=\underset{x\to 0}{\mathop{\lim }}(-x)=0 $
$ \underset{x\to {0^{+}}}{\mathop{\lim }}f(x)=\underset{x\to 0}{\mathop{\lim }}(x)=0 $
and $ f(0)=0 $
So, f(x) is continuous at x = 0 Also, f(x) is continuous for all other values of x.
Hence, f(x) is continuous everywhere.
BETA
