If `f(x)={(1-cosx)/(xsinx),x!=0 and 1/2,x=0` 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

Explanation

To determine the continuity and differentiability of the function \( f(x) = \frac{1 - \cos x}{x \sin x} \) for \( x \neq 0 \) and \( f(0) = \frac{1}{2} \), we will follow these steps:

### Step 1: Check Continuity at \( x = 0 \)

To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0 and compare it to \( f(0) \).

\[
\lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{1 - \cos x}{x \sin x}
...