If f(x) = {((1 - cos x)/(x sin x), x ≠ 0)(1/2, x = 0) then at x = 0,

Question

If \(\text{f(x)} = \begin{cases} \frac{1-cos\,x}{x\,sin\,x} &, x \neq 0 \\ \frac{1}{2} &, x = 0 \end{cases},\) then at x = 0, f(x) is

A. Continuous and differentiable

B. Differentiable but not continuous

C. Continuous but not differentiable

D. Neither continuous nor differentiable

Answer 1

Correct answer is A.

Checking continuity and differentiability at x = 0,

LHL:

LHL = f(x = 0)

Hence, f is continuous at x = 0.

LHD at x = 0,

∵ LHD = RHD = f(0)

∴ f(x) is differentiable at x = 0.