Prove that f(x) = {(sin 3x/x, when x ≠ 0),(1, when x = 0), is discontinuous at x = 0.

Question

Prove that \( f(x) = \begin{cases} sin3x/x, & \quad \text{when } x \text{≠ 0}\\ 1, & \quad \text{when } x \text{ = 0 } \end{cases} \) is discontinuous at x = 0.

Answer 1

\( f(x) = \begin{cases} sin3x/x, & \quad \text{when } x \text{≠ 0}\\ 1, & \quad \text{when } x \text{ = 0 } \end{cases} \)

Consider left hand limit at x = 0

Here the value of function at x = 0 is f(x) = 1 which means f(0) = 1

So

Therefore, f(x) is discontinuous at x = 0.