let f(x)=|x - 9|. evaluate the following limits. lim x→9− f(x)-f(9)/x - 9 = lim x→9+ f(x)-f(9)/x - 9 = thus…

let f(x)=|x - 9|. evaluate the following limits. lim x→9− f(x)-f(9)/x - 9 = lim x→9+ f(x)-f(9)/x - 9 = thus the function f(x) is not differentiable at 9. submit answer next item

let f(x)=|x - 9|. evaluate the following limits. lim x→9− f(x)-f(9)/x - 9 = lim x→9+ f(x)-f(9)/x - 9 = thus the function f(x) is not differentiable at 9. submit answer next item

Answer

Explanation:

Step1: First - find (f(9))

Since (f(x)=|x - 9|), then (f(9)=|9 - 9|=0).

Step2: Calculate the left - hand limit ((x\to9^{-}))

When (x\to9^{-}), (x<9), so (f(x)=|x - 9|=9 - x). Then (\lim_{x\to9^{-}}\frac{f(x)-f(9)}{x - 9}=\lim_{x\to9^{-}}\frac{9 - x-0}{x - 9}=\lim_{x\to9^{-}}\frac{-(x - 9)}{x - 9}=- 1).

Step3: Calculate the right - hand limit ((x\to9^{+}))

When (x\to9^{+}), (x>9), so (f(x)=|x - 9|=x - 9). Then (\lim_{x\to9^{+}}\frac{f(x)-f(9)}{x - 9}=\lim_{x\to9^{+}}\frac{x - 9-0}{x - 9}=1).

Answer:

(\lim_{x\to9^{-}}\frac{f(x)-f(9)}{x - 9}=-1) (\lim_{x\to9^{+}}\frac{f(x)-f(9)}{x - 9}=1)