Question

determine whether f is differentiable at x = 0 by considering lim h→0 (f(0 + h)-f(0))/h. f(x)=18 - |x| choose the correct answer below. a. the function f is differentiable at x = 0 because both the left - and right - hand limits of the difference quotient exist at x = 0. b. the function f is differentiable at x = 0 because the graph has a sharp corner at x = 0. c. the function f is not differentiable at x = 0 because the left - and right - hand limits of the difference quotient exist at x = 0, but are not equal. d. the function f is not differentiable at x = 0 because the left - and right - hand limits of the difference quotient do not exist at x = 0.

Explanation:

Step1: Find f(0)

Given \(f(x)=18 - |x|\), then \(f(0)=18-|0| = 18\).

Step2: Find the left - hand limit of the difference quotient

For \(h\to0^{-}\), \(f(0 + h)=f(h)=18-|h|=18 + h\) (since \(h\lt0\) when \(h\to0^{-}\)).
The left - hand limit \(\lim_{h\to0^{-}}\frac{f(0 + h)-f(0)}{h}=\lim_{h\to0^{-}}\frac{(18 + h)-18}{h}=\lim_{h\to0^{-}}\frac{h}{h}=1\).

Step3: Find the right - hand limit of the difference quotient

For \(h\to0^{+}\), \(f(0 + h)=f(h)=18-|h|=18 - h\) (since \(h\gt0\) when \(h\to0^{+}\)).
The right - hand limit \(\lim_{h\to0^{+}}\frac{f(0 + h)-f(0)}{h}=\lim_{h\to0^{+}}\frac{(18 - h)-18}{h}=\lim_{h\to0^{+}}\frac{-h}{h}=- 1\).

Step4: Determine differentiability

Since \(\lim_{h\to0^{-}}\frac{f(0 + h)-f(0)}{h}=1\) and \(\lim_{h\to0^{+}}\frac{f(0 + h)-f(0)}{h}=-1\), the left - hand and right - hand limits of the difference quotient exist at \(x = 0\) but are not equal.

Answer:

C. The function \(f\) is not differentiable at \(x = 0\) because the left - and right - hand limits of the difference quotient exist at \(x = 0\), but are not equal.