Let `f(x)={(1-cos4x)/(x^2a,),ifx<0(sqrt(x))/(sqrt(16+sqrt(x))-4),ifx >0,ifx=0` Determine the value of `a` so that `f(x)` is continuous at `x=0.`

Explanation

To determine the value of \( a \) so that the function \( f(x) \) is continuous at \( x = 0 \), we need to ensure that the left-hand limit and the right-hand limit at \( x = 0 \) are equal to \( f(0) \).

The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
\frac{1 - \cos(4x)}{x^2 a} & \text{if } x < 0 \\
\frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4} & \text{if } x > 0 \\
...