Chapter 1.

1.1 Problem Statement

{2,4,6,8}
3*$a
6*$a

Use l'Hôpital's Rule to evaluate the limit or state that l'Hôpital's Rule does not apply.

1.2 Step 1

L'Hôpital's Rule states that for functions f(x) and g(x) that are differentiable on an open interval containing x = a, and if f(a) = g(a) = 0, then

if the limit on the right exists or is infinite.

Question Sequence

Question 1.1

To determine whether l'Hôpital's Rule applies to the given problem, evaluate both the numerator and denominator at x = 0.

At x = 0, $a·x3 = 1Wh3cvJ2xF4=.

At x = 0, sin(x) - x = 1Wh3cvJ2xF4=.

Correct.
Incorrect.

Question 1.2

Thus at x = 0, PMOdEMjAlCTr7j/lZi0uQA== an indeterminate form, and since both the numerator and denominator mGbLW+WbgnI0NoznNWVmYQ== differentiable on an open interval containing x = 0, l'Hôpital's Rule bkh5meGQxlYkbqPl4svZiPW4EJ/vGACM.

Correct.
Incorrect.

1.3 Step 2

In order to apply l'Hôpital's Rule on the given limit, determine the derivatives of the numerator and denomintor.

Question 1.3

$a·x3 =j8gjfTtT5bc=·x2

sin(x) - x = XV+Y/un20gdZycFkIAUKN7fBd5V6gRZjT8rpUjhz4oFWLrmO - 1

Correct.
Incorrect.

1.4 Step 3

Apply l'Hôpital's Rule on the given limit.

1.5 Step 4

To determine whether l'Hôpital's Rule applies to , follow the same procedure as in Step 1 by evaluating both the numerator and denominator at x = 0.

Question Sequence

Question 1.4

At x = 0, $a3·x2 = 1Wh3cvJ2xF4=.

At x = 0, cos(x) - 1 = 1Wh3cvJ2xF4=.

Correct.
Incorrect.

Question 1.5

Thus at x = 0, PMOdEMjAlCTr7j/lZi0uQA== an indeterminate form, and since both the numerator and denominator mGbLW+WbgnI0NoznNWVmYQ== differentiable on an open interval containing x = 0, l'Hôpital's Rule bkh5meGQxlYkbqPl4svZiPW4EJ/vGACM.

Correct.
Incorrect.

1.6 Step 5

Question 1.6

In order to apply l'Hôpital's Rule a second time, determine the derivatives of the numerator and denominator.

$a3·x2 = jff51sSOOq4=·x

cos(x) - 1 = kGPYYSI/yYqGToY0eoWcWJNStr8C+qKR0E6C1JC3wX9QMTW/

Correct.
Incorrect.