Let's start by evaluating the function. The function is f(x) = ∣x−9∣. When we plug 9 into this function, f(9) = ∣9-9∣ = 0.
Next, we want to find the limit of the function as x approaches 9 from the left (denoted as x→9-) and from the right (denoted as x→9+).
Let's start with the left. When x approaches 9 from the left side, that means we are dealing with values of x that are just slightly less than 9. In this case, since x is less than 9, (x-9) is negative. So, the absolute function ∣x−9∣ becomes -(x-9), which simplifies to 9-x.
So now we'll find the limit as x→9- of [f(x) - f(9)] / (x - 9), which simplifies to [(9-x) - 0] / (x - 9) = (9-x) / (x - 9).
This is a standard limit problem at this point, which results in -1. So, the left-side limit of the function f(x) at x = 9 is -1.
Now let's calculate the limit from the right. When x approaches 9 from the right, we are looking at values of x just slightly greater than 9. Therefore, (x-9) is positive, so ∣x−9∣ is just (x-9).
Again, we find the limit as x→9+ of [f(x) - f(9)] / (x - 9), which simplifies to [(x-9) - 0] / (x - 9) = (x-9) / (x - 9).
This limit simplifies to 1. Therefore, the right-side limit of the function f(x) at x = 9 is 1.
To summarize, the left-side limit of the function at x = 9 is -1, and the right-side limit at x = 9 is 1. Since the two one-sided limits are not equal, the function is not differentiable at x = 9.