Math / Calculus

QuestionNYA Module 7: Problem 11
For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second enter the numerical value, and in the third case answer DNE. To discourage blind guessing, this problem is graded on the following scale $\begin{array}{l} 0-9 \text { correct = } 0 \\ 10-13 \text { correct }=.3 \\ 14-16 \text { correct }=.5 \\ 17-19 \text { correct }=.7 \end{array}$
Note that l'Hospital's rule (in some form) may ONLY be applied to indeterminate forms.
1. $\infty^{-\infty}$
2. $\infty^{-0}$
3. $\infty^{\infty}$
4. $\infty^{1}$
5. $\pi^{\infty}$
6. $1^{\infty}$ $7.1^{0}$
8. $\infty^{0}$
9. $\pi^{-\infty}$ $10 . \infty \cdot \infty$
11. $\frac{0}{\infty}$
12. $\frac{1}{-\infty}$ $13.0 \cdot \infty$
14. $\frac{\infty}{0}$
15. $1^{-\infty}$ $16.1 \cdot \infty$
17. $\infty-\infty$ $18.0^{0}$ $\square$ $19.0^{\infty}$ $\square$ 20. $0^{-\infty}$

Studdy Solution

STEP 1

1. We are determining the nature of limits involving various forms.
2. The forms can be classified as indeterminate (IND), having a fixed finite value, or never having a fixed finite value (DNE).
3. We will use known mathematical principles and limit properties to classify each form.

STEP 2

1. Analyze each form to determine if it is indeterminate.
2. Determine if the form always has a fixed finite value.
3. Determine if the form never has a fixed finite value.

STEP 3

Analyze the form $\infty^{-\infty}$ :
- This form is indeterminate because the base is infinite and the exponent is negative infinite, creating a conflict in determining the limit.
Result: IND

STEP 4

Analyze the form $\infty^{-0}$ :
- This form simplifies to $1$ because any number raised to the power of zero is $1$ , even if the base is infinite.
Result: 1

STEP 5

Analyze the form $\infty^{\infty}$ :
- This form is not indeterminate; it always results in infinity because an infinite base raised to any positive power is infinite.
Result: DNE

STEP 6

Analyze the form $\infty^{1}$ :
- This form always results in infinity because any number raised to the power of one is itself.
Result: DNE

STEP 7

Analyze the form $\pi^{\infty}$ :
- This form results in infinity because a positive number greater than one raised to an infinite power is infinite.
Result: DNE

STEP 8

Analyze the form $1^{\infty}$ :
- This form is indeterminate because the base is one, but the exponent is infinite, leading to uncertainty in the limit.
Result: IND

STEP 9

Analyze the form $1^{0}$ :
- This form always results in $1$ because any number raised to the power of zero is $1$ .
Result: 1

STEP 10

Analyze the form $\infty^{0}$ :
- This form is indeterminate because the base is infinite and the exponent is zero, leading to uncertainty in the limit.
Result: IND

STEP 11

Analyze the form $\pi^{-\infty}$ :
- This form results in zero because a positive number greater than one raised to a negative infinite power approaches zero.
Result: 0

STEP 12

Analyze the form $\infty \cdot \infty$ :
- This form results in infinity because multiplying two infinite quantities results in infinity.
Result: DNE

STEP 13

Analyze the form $\frac{0}{\infty}$ :
- This form results in zero because zero divided by any infinite quantity approaches zero.
Result: 0

STEP 14

Analyze the form $\frac{1}{-\infty}$ :
- This form results in zero because a finite number divided by an infinite quantity approaches zero.
Result: 0

STEP 15

Analyze the form $0 \cdot \infty$ :
- This form is indeterminate because multiplying zero by an infinite quantity leads to uncertainty in the limit.
Result: IND

STEP 16

Analyze the form $\frac{\infty}{0}$ :
- This form results in infinity because dividing an infinite quantity by zero approaches infinity.
Result: DNE

STEP 17

Analyze the form $1^{-\infty}$ :
- This form results in one because any number raised to the power of zero is one, even with a negative infinite exponent.
Result: 1

STEP 18

Analyze the form $1 \cdot \infty$ :
- This form results in infinity because multiplying one by an infinite quantity results in infinity.
Result: DNE

STEP 19

Analyze the form $\infty-\infty$ :
- This form is indeterminate because subtracting two infinite quantities leads to uncertainty in the limit.
Result: IND

STEP 20

Analyze the form $0^{0}$ :
- This form is indeterminate because zero raised to the power of zero is a classic indeterminate form.
Result: IND

STEP 21

Analyze the form $0^{\infty}$ :
- This form results in zero because zero raised to any positive power is zero.
Result: 0

STEP 22

Analyze the form $0^{-\infty}$ :
- This form results in infinity because zero raised to a negative infinite power approaches infinity.
Result: DNE

Was this helpful?

Studdy Solution

STEP 1

1. We are determining the nature of limits involving various forms.2. The forms can be classified as indeterminate (IND), having a fixed finite value, or never having a fixed finite value (DNE).3. We will use known mathematical principles and limit properties to classify each form.

STEP 2

1. Analyze each form to determine if it is indeterminate.2. Determine if the form always has a fixed finite value.3. Determine if the form never has a fixed finite value.

STEP 3

Analyze the form ∞−∞\infty^{-\infty}∞−∞:- This form is indeterminate because the base is infinite and the exponent is negative infinite, creating a conflict in determining the limit.Result: IND

STEP 4

Analyze the form ∞−0\infty^{-0}∞−0:- This form simplifies to 111 because any number raised to the power of zero is 111, even if the base is infinite.Result: 1

STEP 5

Analyze the form ∞∞\infty^{\infty}∞∞:- This form is not indeterminate; it always results in infinity because an infinite base raised to any positive power is infinite.Result: DNE

STEP 6

Analyze the form ∞1\infty^{1}∞1:- This form always results in infinity because any number raised to the power of one is itself.Result: DNE

STEP 7

Analyze the form π∞\pi^{\infty}π∞:- This form results in infinity because a positive number greater than one raised to an infinite power is infinite.Result: DNE

STEP 8

Analyze the form 1∞1^{\infty}1∞:- This form is indeterminate because the base is one, but the exponent is infinite, leading to uncertainty in the limit.Result: IND

STEP 9

Analyze the form 101^{0}10:- This form always results in 111 because any number raised to the power of zero is 111.Result: 1

STEP 10

Analyze the form ∞0\infty^{0}∞0:- This form is indeterminate because the base is infinite and the exponent is zero, leading to uncertainty in the limit.Result: IND

STEP 11

Analyze the form π−∞\pi^{-\infty}π−∞:- This form results in zero because a positive number greater than one raised to a negative infinite power approaches zero.Result: 0

STEP 12

Analyze the form ∞⋅∞\infty \cdot \infty∞⋅∞:- This form results in infinity because multiplying two infinite quantities results in infinity.Result: DNE

STEP 13

Analyze the form 0∞\frac{0}{\infty}∞0​:- This form results in zero because zero divided by any infinite quantity approaches zero.Result: 0

STEP 14

Analyze the form 1−∞\frac{1}{-\infty}−∞1​:- This form results in zero because a finite number divided by an infinite quantity approaches zero.Result: 0

STEP 15

Analyze the form 0⋅∞0 \cdot \infty0⋅∞:- This form is indeterminate because multiplying zero by an infinite quantity leads to uncertainty in the limit.Result: IND

STEP 16

Analyze the form ∞0\frac{\infty}{0}0∞​:- This form results in infinity because dividing an infinite quantity by zero approaches infinity.Result: DNE

STEP 17

Analyze the form 1−∞1^{-\infty}1−∞:- This form results in one because any number raised to the power of zero is one, even with a negative infinite exponent.Result: 1

STEP 18

Analyze the form 1⋅∞1 \cdot \infty1⋅∞:- This form results in infinity because multiplying one by an infinite quantity results in infinity.Result: DNE

STEP 19

Analyze the form ∞−∞\infty-\infty∞−∞:- This form is indeterminate because subtracting two infinite quantities leads to uncertainty in the limit.Result: IND

STEP 20

Analyze the form 000^{0}00:- This form is indeterminate because zero raised to the power of zero is a classic indeterminate form.Result: IND

STEP 21

Analyze the form 0∞0^{\infty}0∞:- This form results in zero because zero raised to any positive power is zero.Result: 0

STEP 22

Analyze the form 0−∞0^{-\infty}0−∞:- This form results in infinity because zero raised to a negative infinite power approaches infinity.Result: DNE

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are determining the nature of limits involving various forms.
2. The forms can be classified as indeterminate (IND), having a fixed finite value, or never having a fixed finite value (DNE).
3. We will use known mathematical principles and limit properties to classify each form.

1. Analyze each form to determine if it is indeterminate.
2. Determine if the form always has a fixed finite value.
3. Determine if the form never has a fixed finite value.

Analyze the form $\infty^{-\infty}$ :
- This form is indeterminate because the base is infinite and the exponent is negative infinite, creating a conflict in determining the limit.
Result: IND

Analyze the form $\infty^{-0}$ :
- This form simplifies to $1$ because any number raised to the power of zero is $1$ , even if the base is infinite.
Result: 1

Analyze the form $\infty^{\infty}$ :
- This form is not indeterminate; it always results in infinity because an infinite base raised to any positive power is infinite.
Result: DNE

Analyze the form $\infty^{1}$ :
- This form always results in infinity because any number raised to the power of one is itself.
Result: DNE

Analyze the form $\pi^{\infty}$ :
- This form results in infinity because a positive number greater than one raised to an infinite power is infinite.
Result: DNE

Analyze the form $1^{\infty}$ :
- This form is indeterminate because the base is one, but the exponent is infinite, leading to uncertainty in the limit.
Result: IND

Analyze the form $1^{0}$ :
- This form always results in $1$ because any number raised to the power of zero is $1$ .
Result: 1

Analyze the form $\infty^{0}$ :
- This form is indeterminate because the base is infinite and the exponent is zero, leading to uncertainty in the limit.
Result: IND

Analyze the form $\pi^{-\infty}$ :
- This form results in zero because a positive number greater than one raised to a negative infinite power approaches zero.
Result: 0

Analyze the form $\infty \cdot \infty$ :
- This form results in infinity because multiplying two infinite quantities results in infinity.
Result: DNE

Analyze the form $\frac{0}{\infty}$ :
- This form results in zero because zero divided by any infinite quantity approaches zero.
Result: 0

Analyze the form $\frac{1}{-\infty}$ :
- This form results in zero because a finite number divided by an infinite quantity approaches zero.
Result: 0

Analyze the form $0 \cdot \infty$ :
- This form is indeterminate because multiplying zero by an infinite quantity leads to uncertainty in the limit.
Result: IND

Analyze the form $\frac{\infty}{0}$ :
- This form results in infinity because dividing an infinite quantity by zero approaches infinity.
Result: DNE

Analyze the form $1^{-\infty}$ :
- This form results in one because any number raised to the power of zero is one, even with a negative infinite exponent.
Result: 1

Analyze the form $1 \cdot \infty$ :
- This form results in infinity because multiplying one by an infinite quantity results in infinity.
Result: DNE

Analyze the form $\infty-\infty$ :
- This form is indeterminate because subtracting two infinite quantities leads to uncertainty in the limit.
Result: IND

Analyze the form $0^{0}$ :
- This form is indeterminate because zero raised to the power of zero is a classic indeterminate form.
Result: IND

Analyze the form $0^{\infty}$ :
- This form results in zero because zero raised to any positive power is zero.
Result: 0

Analyze the form $0^{-\infty}$ :
- This form results in infinity because zero raised to a negative infinite power approaches infinity.
Result: DNE