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 case enter the numerical value, and in the third case answer DNE. Here are examples of the three cases:
$\bullet$ Type $\frac{0}{0}$ means a limit of the form $\displaystyle \lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ where $\displaystyle \lim_{x \rightarrow a} f(x) = 0$ and $\displaystyle \lim_{x \rightarrow a} g(x) = 0$. Since this is an indeterminate form, the answer is IND.
$\bullet$ Type $\frac{0}{1}$ means a limit of the form $\displaystyle \lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ where $\displaystyle \lim_{x \rightarrow a} f(x) = 0$ and $\displaystyle \lim_{x \rightarrow a} g(x) = 1$. Since this limit is always 0, the answer is 0.
$\bullet$ Type $\frac{1}{0}$ means a limit of the form $\displaystyle \lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ where $\displaystyle \lim_{x \rightarrow a} f(x) = 1$ and $\displaystyle \lim_{x \rightarrow a} g(x) = 0$. Since $\frac{f(x)}{g(x) }$ never converges to a finite value, the answer is DNE.
Note that l'Hospital's rule (in some form) may ONLY be applied to indeterminate forms.
1. $1^\infty$
2. $\infty^{-e}$
3. $1\cdot\infty$
4. $\infty\cdot\infty$
5. $\pi^{-\infty}$
6. $\infty^0$
7. $\infty^1$
8. $\frac{1}{-\infty}$
9. $0^0$
10. $\infty^{-\infty}$
11. $0^\infty$
12. $\infty -\infty$
13. $\pi^\infty$
14. $1^0$
15. $\frac{0}{\infty}$
16. $\infty^\infty$
17. $\frac{\infty}{0}$
18. $0^{-\infty}$
19. $1^{-\infty}$
20. $0\cdot\infty$