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Chapter 2: Q. 71 (page 185)

In Exercises 69-80, determine whether or not fis continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

f(x)=x2-4,x=2

Short Answer

Expert verified

The function is continuous at x=2but not differentiable and the graph is

Step by step solution

01

Step 1. Given information

Given functionf(x)=x2-4,x=2

02

Calculate the LHL and RHL and calculate

Calculating, we get

limx鈫�2-f(x)=limx鈫�2-4-x2=4-22=0limx鈫�2+f(x)=limx鈫�2+x2-4=22-4=0f(2)=22-4=0

So the function is continuous at x=2

03

Checking for differentiability

Checking, we get

limx鈫�2-f(x)-f(2)x-2=limx鈫�2-4-x2-0x-2limx鈫�2-f(x)-f(2)x-2=limx鈫�2--x2-4x-2=limx鈫�2--(x+2)(x-2)x-2=limx鈫�2-(-(x+2)=-(2+2)=-4

The function is not differentiable atx=2

04

Checking the graph

The graph is

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