`f(x)= {{:((|x-4|)/(2(x-4)), if x ne 4),(0,if x = 4):}` check limit at `x = 4` is.

Explanation

To solve the problem of finding the limit of the function \( f(x) = \frac{|x-4|}{2(x-4)} \) as \( x \) approaches 4, we will evaluate the left-hand limit and the right-hand limit separately.

### Step 1: Define the function
The function is defined as:
\[
f(x) =
\begin{cases}
\frac{|x-4|}{2(x-4)} & \text{if } x \neq 4 \\
...