Final answer:
To prove that the function f(x) = |x-1| is not differentiable at x = 1 using the limit definition of the derivative, we need to show that the left-hand limit and the right-hand limit of the difference quotient do not equal each other. By finding the left-hand limit and the right-hand limit and showing that they are not equal, we can conclude that the function is not differentiable at x = 1.
Step-by-step explanation:
To prove that the function f(x) = |x-1| is not differentiable at x = 1 using the limit definition of the derivative, we need to show that the left-hand limit and the right-hand limit of the difference quotient do not equal each other.
Step 1:
Start by finding the left-hand limit:
limh→0-[f(1+h) - f(1)] / h
Substitute the function f(x):
limh→0-[|1+h-1| - |1-1|] / h
Simplify:
limh→0-[|h|] / h
As h approaches 0 from the left, |h| / h approaches -1.
Step 2:
Next, find the right-hand limit:
limh→0+[f(1+h) - f(1)] / h
Substitute the function f(x):
limh→0+[|1+h-1| - |1-1|] / h
Simplify:
limh→0+[|h|] / h
As h approaches 0 from the right, |h| / h approaches 1.
Step 3:
Since the left-hand limit (-1) does not equal the right-hand limit (1), the function is not differentiable at x = 1.