Question

Find the limit, use l’Hopital’s Rule where appropriate. If there is a more elementary method, consider using it

(a) lim x→0+ (x+sin(2x)) / ( tan(5x)) .

(b) limx→∞ (x^2+3x+1) /(x ln x) .

(c) limx→∞ [ xe^(2/x) − x]

(d) limx→∞ [ln(3x + 2) − ln(5x − 3)].

(e) lim x→0− ( (1/x) − csc x).

(f) limx→∞ (1 − (5/ x^2))^ (
x^ 2) .

(g) limx→∞ (e^(x) + 1)^(− 2/x) .

Answer 1

(a)

Step 1:

The expression is .

Rewrite the expression as:.

Apply sum and difference rule in limits .

Apply L-Hospital rule for

.

Apply formula .

.

Solution:

.

Answer 2

(b)

Step 1:

The expression is .

Rewrite the expression as .

Apply sum and difference rule in limits .

Apply L-Hospital rule.

.

Solution:

.

Answer 3

(c)

Step 1:

The expression is .

Rewrite the expression as .

Apply L-Hospital rule:

.

Solution:

.

Answer 4

(d)

Step 1:

The expression is .

Apply quotient rule of logarithms:.

.

.

Solution:

.

Answer 5

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:

.

Answer 6

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:

.

Answer 7

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:

.