Step 1:

(e)

The limit of the function is .

Rewrite the expression.

L-Hospital rule:

Suppose or , then .

Apply L-Hospital rule.

Substitute in the above expression.

.

Step 2:

Again apply L-Hospital rule. \ \

Substitute in the above expression.

Solution:

.

Step 1:

(f)

The limit of the function is .

Consider .

Apply logarithm on each side.

Rewrite the expression.

Apply limits on each side.

Step 2:

L-Hospital rule:

Suppose or , then

Apply L-Hospital rule. \ \

As then .

.

Solution:

.

Step 1:

(g) 

The limit of the function is .

Consider .

Apply logarithm on each side.

Apply limits on each side.

Step 2:

L-Hospital rule:

Suppose or , then

Apply L-Hospital rule.

Multiply numerator and denominator by .

As then .

.

Solution:

.