12.2.2 L'Hôpital's Rule and Indeterminate Differences Flashcards
1
Q
L’Hôpital’s Rule and Indeterminate Differences
A
- 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
2
Q
note
A
- 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.
3
Q
Evaluatelimx→∞ (4√x^4 + x^3 – x).
A
1/4
4
Q
Evaluate limx→∞(3√x^3+x^2−x)
A
1/3
5
Q
Evaluate limx→0 (1/x – 1/ln(1 + x)).
A
−1/2
6
Q
Evaluate limx→2 (1/x − 2 – 1/ln(x − 1))
A
−1/2
7
Q
Evaluate limc→1(2cc2+c−2−1c−1).
A
The limit does not exist.
8
Q
Evaluate limx→∞ (√x + 2 – √x).
A
0
9
Q
Evaluate limx→0 (1x – cot x)
A
0
10
Q
Evaluatelimx→∞(x5−1000x4).
A
∞
11
Q
Evaluate limx→∞ (√9x2 + 2x − 3x).
A
1/3
12
Q
Evaluate limx→0 (1x – 1sinx).
A
0
13
Q
Evaluate limx→0 ⎛⎜⎝1ln(x + √1 + x2) – 1ln(1 + x)⎞⎟⎠
A
−1/2
14
Q
Evaluate limx→∞(√x2+3x−x).
A
3/2