Step 1: Define the function

The given function is $f(x) = |x-5|$. We can rewrite this as a piecewise function:

$$f(x) = \begin{cases} x-5, & \text{if } x \geq 5 \\ -(x-5), & \text{if } x < 5 \end{cases}$$

Step 2: Check the left-hand derivative

The left-hand derivative at $x=5$ is given by:

$$LHD = \lim_{h \to 0^-} \frac{f(5+h) - f(5)}{h}$$

Since $h \to 0^-$, $5+h < 5$, so we use the second part of the piecewise function:

$$LHD = \lim_{h \to 0^-} \frac{-(5+h-5) - |5-5|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$$

Step 3: Check the right-hand derivative

The right-hand derivative at $x=5$ is given by:

$$RHD = \lim_{h \to 0^+} \frac{f(5+h) - f(5)}{h}$$

Since $h \to 0^+$, $5+h > 5$, so we use the first part of the piecewise function:

$$RHD = \lim_{h \to 0^+} \frac{(5+h-5) - |5-5|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1$$

Step 4: Compare LHD and RHD

We have $LHD = -1$ and $RHD = 1$. Since $LHD \neq RHD$, the function is not differentiable at $x=5$.