Step 1:

The expression is .

definition:

means that for every , there exists a , such that for every , the expression  implies .

Consider .

Adding each side by .

As ,

Step 2:

Consider .

As ,

.

Observe the relation between and  .

Hence,

Solution:

.

(a)

Step 1:

The limit of the function is .

definition of limit :

,

if for every number of , there exists a such that whenever .

From the definition, if then .

Step 2:

When .

Consider

\ \

Graph :

Graph of the function :

Observe the graph:

The function lies in the interval .

Step 3:

Consider .

.

The function lies in the interval .

Compare with .

Here value should be a smaller positive number.

Therefore .

When .

Consider

. \ \

Graph :

Graph of the function :

Observe the graph:

The function lies in the interval .

Consider .

.

The function lies in the interval .

.

Compare with .

Here value should be a smaller positive number.

Therefore .

When then .

When then .