12.2.2 L'Hôpital's Rule and Indeterminate Differences

Question 1

L’Hôpital’s Rule and Indeterminate Differences

Answer 1

• Some indeterminate forms have to be transformed before you can apply L’Hôpital’s rule.
• Look for a common denominator or a clever way of factoring to transform an indeterminate difference into an indeterminate quotient to which you can apply L’Hôpital’s rule

Question 2

note

Answer 2

• This is an example of an indeterminate difference that you can transform by finding a common denominator.
• Once you have expressed the limit as quotient, it produces the standard indeterminate form 0/0.
• A second application of L’Hôpital’s rule is needed since the limit produces an indeterminate form again.
• This limit produces an indeterminate difference, but it’s not obvious how to find a common denominator.
• Try factoring the expression, being very careful when working under the radical.
• Once you have factored out x, you can send it to the
  denominator by finding its reciprocal,
• Now you have a limit that produces the form
  apply L’Hôpital’s rule. , so you can
• The numerator includes a square-root expression, so you’ll have to use the chain rule.
• Cancel common factors and plug in the value to determine the limit.

Question 3

Evaluatelimx→∞ (4√x^4 + x^3 – x).

Answer 3

1/4

Question 4

Evaluate limx→∞(3√x^3+x^2−x)

Answer 4

1/3

Question 5

Evaluate limx→0 (1/x – 1/ln(1 + x)).

Answer 5

−1/2

Question 6

Evaluate limx→2 (1/x − 2 – 1/ln(x − 1))

Answer 6

−1/2

Question 7

Evaluate limc→1(2cc2+c−2−1c−1).

Answer 7

The limit does not exist.

Question 8

Evaluate limx→∞ (√x + 2 – √x).

Answer 8

0

Question 9

Evaluate limx→0 (1x – cot x)

Answer 9

0

Question 10

Evaluatelimx→∞(x5−1000x4).

Answer 10

∞

Question 11

Evaluate limx→∞ (√9x2 + 2x − 3x).

Answer 11

1/3

Question 12

Evaluate limx→0 (1x – 1sinx).

Answer 12

0

Question 13

Evaluate limx→0 ⎛⎜⎝1ln(x + √1 + x2) – 1ln(1 + x)⎞⎟⎠

Answer 13

−1/2

Question 14

Evaluate limx→∞(√x2+3x−x).

Answer 14

3/2