12.1.3 Basic Uses of L'Hôpital's Rule

Question 1

Basic Uses of L’Hôpital’s Rule

Answer 1

• When evaluating a limit, it is always a good idea to plug in the value first. If the result yields an indeterminate form, then use L’Hôpital’s rule.
• As long as the limit is still an indeterminate form, you can reuse L’Hôpital’s rule.
• L’Hôpital’s rule might not produce the right answer if you use it on a limit that does not produce an indeterminate form.

Question 2

note

Answer 2

• Make sure to plug the value –2 in place of x as your first step. Remember, you can only use L’Hôpital’s rule if the limit produces the indeterminate forms or
• Once you have verified that the limit produces an
  indeterminate form, L’Hôpital’s rule works by taking the derivative of the numerator and the derivative of the denominator. These form a new limit that may be easier to evaluate.
• Before using L’Hôpital’s rule, make a guess as to the value of the limit. It’s okay if your guess is wrong—guessing will help you learn the patterns that lead to various answers.
• This is a mean example of a limit because the higher-power terms are not in the front. Terms with higher powers have a greater effect on the limit when x approaches infinity.
• Since –1000 and 103x have relatively low powers, their effect is negligible. The remaining terms are both raised to the 103rd power, so they will cancel each other out and leave only their coefficients. For this reason, 1/2 is a good guess.
• For this example, Professor Burger played the role of a student who applied L’Hôpital’s rule.
• However, if you plug in first, you will see that this limit does not produce an indeterminate form. Evaluating the limit directly will be less work and will give an answer you can be sure of. Using L’Hôpital’s rule incorrectly might result in a wrong answer.

Question 3

Evaluate.

limx→1 x^3 − 1/x^2 − 1

Answer 3

3/2

Question 4

Evaluate lim x→0 2x^3 − 128/16 − 3x^3.

Answer 4

−8

Question 5

Evaluate lim x→∞ 2x^2−7/5x^2+9x+2

Answer 5

2/5

Question 6

Evaluate limx→2 x^3 − 8/x^4 − 16.

Answer 6

3/8

Question 7

Evaluate.

limx→∞ x^1000 + 6000/6 − 3x^1000

Answer 7

−1/3

Question 8

Evaluate lim x→1 x^n − x^m/2x − 2where n>m>1.

Answer 8

n-m/2

Question 9

Evaluate limy→8 3y − 24/64 − y^2

Answer 9

-3/16

Question 10

Evaluate limy→1 2y − 2/1 − y^100

Answer 10

−1/50