Note that l'Hospital's rule 0 1. $$ \infty^{-\infty} $$ IND 2. $$ 0 \cdot \infty $$ IND 3. $$ 1^{\infty} $$ $$ \infty $$ 4.1. $$ \infty $$ IND 5. $$ 0^{0} $$ DNE 6. $$ 0^{-\infty} $$ 0 7. $$ \pi^{-\infty} $$ $$ \infty $$ 8. $$ \pi^{\infty} $$ $$ \infty $$ 9. $$ \infty \cdot \infty $$ 0 10. $$ \infty^{-e} $$ IND 11. $$ \infty^{\infty} $$ IND 12. $$ \infty-\infty $$ $$ D N E $$ 13. $$ \frac{\infty}{0} $$ IND 14. $$ 0^{\infty} $$ 1 15. $$ 1^{0} $$ 0 16. $$ \frac{1}{-\infty} $$ $$ \infty $$ 17. $$ \infty^{1} $$ 0 18. $$ \frac{0}{\infty} $$ IND 19. $$ \infty^{0} $$ 0 20. $$ 1^{-\infty} $$

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To analyze the indeterminate forms and their limits, we can categorize them based on the expressions provided. Here’s a detailed breakdown of each case:

1. **$$ \infty^{-\infty} $$**: This form tends to $$ 0 $$ because as the base approaches infinity and the exponent approaches negative infinity, the overall expression approaches zero. **Result: 0**

2. **$$ 0 \cdot \infty $$**: This is an indeterminate form (IND) because it can yield different results depending on the context. For example, $$ \lim_{x 	o 0} x \cdot \frac{1}{x} = 1 $$ while $$ \lim_{x 	o 0} x \cdot \frac{1}{x^2} = \infty $$. **Result: IND**

3. **$$ 1^{\infty} $$**: This is also an indeterminate form (IND) because it can yield different results depending on the limit. For example, $$ \lim_{x 	o 0} (1 + x)^{\frac{1}{x}} = e $$. **Result: IND**

4. **$$ \infty $$**: This is a straightforward case where the limit approaches infinity. **Result: $$ \infty $$**

5. **$$ 0^{0} $$**: This is considered indeterminate (IND) because it can yield different results depending on the context. For example, $$ \lim_{x 	o 0} x^x = 1 $$. **Result: IND**

6. **$$ 0^{-\infty} $$**: This tends to $$ \infty $$ because a base approaching zero raised to a negative exponent diverges. **Result: $$ \infty $$**

7. **$$ \pi^{-\infty} $$**: This tends to $$ 0 $$ because a positive constant raised to a negative infinite power approaches zero. **Result: 0**

8. **$$ \pi^{\infty} $$**: This tends to $$ \infty $$ because a positive constant raised to an infinite power diverges. **Result: $$ \infty $$**

9. **$$ \infty \cdot \infty $$**: This clearly tends to $$ \infty $$ as both factors are approaching infinity. **Result: $$ \infty $$**

10. **$$ \infty^{-e} $$**: This tends to $$ 0 $$ because a base approaching infinity raised to a negative exponent approaches zero. **Result: 0**

11. **$$ \infty^{\infty} $$**: This tends to $$ \infty $$ because a base approaching infinity raised to an infinite exponent diverges. **Result: $$ \infty $$**

12. **$$ \infty - \infty $$**: This is an indeterminate form (DNE) because it can yield different results depending on the context. **Result: DNE**

13. **$$ \frac{\infty}{0} $$**: This tends to $$ \infty $$ because dividing infinity by a number approaching zero diverges. **Result: $$ \infty $$**

14. **$$ 0^{\infty} $$**: This tends to $$ 0 $$ because a base approaching zero raised to an infinite exponent approaches zero. **Result: 0**

15. **$$ 1^{0} $$**: This is a defined case and equals $$ 1 $$. **Result: 1**

16. **$$ \frac{1}{-\infty} $$**: This tends to $$ 0 $$ because a constant divided by negative infinity approaches zero. **Result: 0**

17. **$$ \infty^{1} $$**: This clearly tends to $$ \infty $$ as the base is approaching infinity. **Result: $$ \infty $$**

18. **$$ \frac{0}{\infty} $$**: This tends to $$ 0 $$ because zero divided by any positive number approaches zero. **Result: 0**

19. **$$ \infty^{0} $$**: This is an indeterminate form (IND) because it can yield different results depending on the context. **Result: IND**

20. **$$ 1^{-\infty} $$**: This tends to $$ 0 $$ because a base of one raised to any negative exponent approaches zero. **Result: 0**

In summary, the results are as follows:

1. 0
2. IND
3. IND
4. $$ \infty $$
5. IND
6. $$ \infty $$
7. 0
8. $$ \infty $$
9. $$ \infty $$
10. 0
11. $$ \infty $$
12. DNE
13. $$ \infty $$
14. 0
15. 1
16. 0
17. $$ \infty $$
18. 0
19. IND
20. 0
Here are the simplified results for each indeterminate form: 1. 0 2. IND 3. IND 4. $$ \infty $$ 5. IND 6. $$ \infty $$ 7. 0 8. $$ \infty $$ 9. $$ \infty $$ 10. 0 11. $$ \infty $$ 12. DNE 13. $$ \infty $$ 14. 0 15. 1 16. 0 17. $$ \infty $$ 18. 0 19. IND 20. 0

Note that l’Hospital’s rule
0 1.
IND 2.
IND 3.
4.1.
IND 5.
DNE 6.
0 7.
8.
9.
0 10.
IND 11.
IND 12.
13.
IND 14.
1 15.
0 16.
17.
0 18.
IND 19.
0 20.

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