find a number δ such that...

Question

 

1) Use the given graph of f(x) = x² to find a number δ such that

if    |x − 1| < δ    then    |x² − 1| <1/2

http://imgur.com/HcmW5yc

 

(Round your answer down to three decimal places.)

δ =?

 

2) Use the given graph of f to find a number δ such that if 

http://imgur.com/qIpDK9r

|x − 1| < δ    then    |f(x) − 1| < 0.2

δ =?

 

Answer 1

Step 1:

(1)

The function is .

Find such that    whenever .

Definition of limit:

Let be a function defined on an open interval containing and let be a real number.

The statement

means that for each , there exists a  such that , then .

Step 2:

  whenever .

We need to establish a connection between   and .

Consider .

Solve the inequality.

Subtract from each side.

Since ,

Compare the above with , then .

Solution:

.

Answer 2

Step 1:

(2)

Find such that    whenever .

Definition of limit:

Let be a function defined on an open interval containing and let be a real number.

The statement

means that for each , there exists a  such that , then .

Step 2:

  whenever .

Consider .

Observe the graph:

Compare and .

Lower value:

Upper value:

The values of are 0.1 and 0.3

Solution:

The values of are 0.1 and 0.3.