find the limit. Use L'Hopital's Rule where appropiate

Question

lim x approaching infinity ((1) / (e^(3x)+1))^(1/(2x))

lim x approaching infinity sqrt(x^2 +4x)-x

lim x approaching infinity (x^2)e^(-x)

Answer 1

Step 1:

The expression is .

Step 2:

L-Hospital rule:

Suppose or , then

Apply L-Hospital rule.

As then .

.

.

Solution:

.

Answer 2

(3)

Step 1:

The limit of the function is .

L'Hospital's Rule :

 If the value of limit is indeterminate form of type  or ,

then .

Step 2:

Since use L'Hospital's Rule.

If we evaluate the above limit, value will be obtained.

Again apply L'Hospital's Rule.

.

Solution :

.

Answer 3

(1)

Step 1:

The expression is .

Consider .

Apply logarithm on each side.

Apply limits on each side as .

.

Step 2:

L-Hospital rule:

Suppose or , then

Apply L-Hospital rule.

Multiply numerator and denominator by .

As then .

.

.

Solution:

.