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

Explanation

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

### Step 1: Define the function and limits
The function is defined as:
\[
f(x) =
\begin{cases}
\frac{1 - \cos(4x)}{x^2} & \text{if } x < 0 \\
...