​ Let f(x) = (x sin 1/x, if x ≠ 0) (0, when x = 0). Then, which of the following is the true statement? A. f(x) is not defined at x = 0

Question

Let f(x) = \(\begin{cases} x\,sin\,1/x, & if\,x\neq0 \\ 0, & when\,x\neq0. \end{cases}\)

x sin 1/x, if x ≠ 0

0, when x = 0

Then, which of the following is the true statement?

A. f(x) is not defined at x = 0

B. \(\lim\limits_{x\to0}\) f(x) does not exist

C. f(x) is continuous at x = 0

D. f(x) is discontinuous at x = 0

Answer 1

Answer is: C. f(x) is continuous at x = 0

Left hand limit = 

Right hand limit =

= 1

As L.H.L = R.H.L

F(x) is continuous.