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Chapter 14: Q. 12 (page 907)

Determine whether fis continuous at each of the following numbers. If it is not, explain why not.

(a)x=-2(b)x=1(c)x=3(d)x=4

Short Answer

Expert verified

The function is continuous at x=-2,4and discontinuous atx=1,3

Step by step solution

01

Part (a) Step 1. Given Information

A graph is given.

We need to find if the function is continuous atx=-2

02

Part(a) Step 2. Explanation

For x=2,f(2)=2It is clear from the graph that,

limx鈫�-2-f(x)=2limx鈫�-2+f(x)=2limx鈫�-2-f(x)=2=limx鈫�-2+f(x)limx鈫�-2f(x)=2

Therefore, continuous limit exists.

03

f(1)=0Part(b) Step 1. Explanation

For x=1

It is clear from the graph that,

limx鈫�1-f(x)=2limx鈫�1+f(x)=2limx鈫�1-f(x)=2=limx鈫�1+f(x)limx鈫�1f(x)=2Butf(1)=0

Therefore, continuous limit does not exist.

04

Part(c) Step 1. Explanation

For x=3,f(3)=-3

It is clear from the graph,

limx鈫�3-f(x)=5limx鈫�3+f(x)=-3

Since,

limx鈫�3-f(x)=5鈮�-3=limx鈫�3+f(x)

Therefore, continuous limit does not exist.

05

Part(d) Step 1. Explanation

For x=4,f(4)=-4

From the graph,

limx鈫�4f(x)=-4

Therefore, continuous limit exists.

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