(a) Write the inequalities |f(x) − L| < epsilon and |x− a| < delta as they pertain to this statement.

Question

. Given the limit statement lim x→−2 (−3x + 1) = 7

(a) Write the inequalities |f(x) − L| < epsilon and |x− a| < delta as they pertain to this statement.

(b) Illustrate the definition of limit by finding a number delta that corresponds to epsilon = 0.06, repeat for epsilon = 0.0001.

Answer 1

(a)

Step 1:

The limit function is .

The function is .

Definition of the limit :

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

The statement is , means that for each there exists a such that

if , then .

Step 2:

From the definition :

Solution:

(a) and .

Answer 2

(b)

Step 1:

The limit function is .

(delta) value for :

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

if , then .

Step 2:

  whenever .

We need to establish a connection between   and .

Consider .

Since ,

Compare the above with , then .

Answer 3

Contd.....

Step 3:

For :

Find such that    whenever .

  whenever .

We need to establish a connection between   and .

Consider

Since ,

Compare the above with , then .

Solution:

(b) .

     .