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Chapter 1: Q. 80 (page 89)

Prove that for all x within 0.01 of the value x = 1, the quantity x-12 is within the interval (0, 0.0001). What does this have to do withlimx鈫�1x-12?

Short Answer

Expert verified

The given statement is proved. The value of the limitlimx鈫�1x-12=0.

Step by step solution

01

Step 1. Given Information.  

The given quantity isx-12.

02

Step 2. Prove.  

Let the function isf(x)=x-12.

Take the limit of the above function as x鈫�0.01:

limx鈫�0.01f(x)=limx鈫�0.01x-12limx鈫�0.01f(x)=0.01-12limx鈫�0.01f(x)=-0.992limx鈫�0.01f(x)=0.9801

Now, take the limit of the above function as x鈫�1:

limx鈫�1f(x)=limx鈫�1x-12limx鈫�1f(x)=1-12limx鈫�1f(x)=0

Therefore, the quantityx-12is within the interval0,0.0001.

03

Step 3. Finding the limit.

Let's find the limit.

limx鈫�1x-12=1-12=0

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