Question

Let a,b, and c denote complex constants. Then use definition (2), Sec. 15, of limit to show that

Answer 1

Definition of the limit:

The limit of as z approaches to is a number

That is,

This means that, for every there exists a positive number such that whenever

(a)

The objective is to prove that by using the definition of limit.

Where, a, and b are complex constants.

Assume that,

Assume and .

If then for any choice of .

If ,

So,

Choose then,

For any z and every positive number whenever

Therefore,

(b)

The objective is to prove that by using the definition of limit.

Where, c is complex constant.

Assume that,

Assume be any positive integer for a given , such that

Which implies

So,

Now restrict and observe that

Choose

For any z and every positive number whenever

Therefore,

(c)

The objective is to prove that by using the definition of limit.

Where, c is complex constant, and

Assume and Then and

Here, means and

Assume that,

Consider,

So,

Choose then

For any z and every positive number whenever

Therefore,