Limits of Indeterminate Form? I understand that $1^\infty$ is an indeterminate form.  But what would
$$\frac{1^\infty}{1^\infty}$$
Be considered?  Am I able to say that this is just 1?  
 A: Suppose that $\displaystyle\lim_{x\to c} f(x) = 1$ and $\displaystyle\lim_{x\to c} g(x) = \infty$. This information is not enough to determine $\displaystyle\lim_{x\to c} f(x)^{g(x)}$, or even if that limit exists. The limit can be any nonnegative number, or even infinity. It is for this reason that we say that $1^\infty$ is an indeterminate form.
Analogously, to say that $\frac{1^\infty}{1^\infty}$ is an indeterminate form must mean that assuming
$$\lim_{x\to c} f_1(x) = 1, \qquad \lim_{x\to c} g_1(x) = \infty,$$
$$\lim_{x\to c} f_2(x) = 1, \qquad \displaystyle\lim_{x\to c} g_2(x) = \infty,$$
is not enough to determine the value of
$$\lim_{x\to c} \frac{f_1(x)^{g_1(x)}}{f_2(x)^{g_2(x)}}.$$
This is the case. To see this, consider that, because $1^\infty$ is an indeterminate form, by choosing the $f_i$ and $g_i$ functions appropriately, we can make both the numerator and denominator have whatever nonnegative limit we want. So, the whole fraction can have any nonnegative limit value. Therefore, $\frac{1^\infty}{1^\infty}$ is an indeterminate form.
A: No, this is also an indeterminate form. It is not equal to $1$. Depending on the origin of $1^{\infty}$, it may be sensical for it to be $1$ or $\infty$ or anything in between, and as long as you don't take the origin of the limit into account, you have no idea how a fraction like
$$\frac{1^{\infty}}{1^{\infty}}$$
behaves.
It is similar to an indeterminate expression like $\frac{0}{0}$. Is this equal to $1$? No. Again, it depends, and without any further information, you cannot manipulate it in any way.
To show you what I mean, here's an example, where $\frac{0}{0}$ informally turns out to be $7$:
$$\frac{0}{0}=\frac{\lim_{x\rightarrow\infty} 7/x}{\lim_{x\rightarrow\infty} 1/x} = \lim_{x\rightarrow\infty} \frac{7/x}{1/x} = \lim_{x\rightarrow\infty} 7 = 7.$$
On one hand, it looked like I had an indeterminate expression, but on the other hand, it really originated from the limit of a fraction, whose limit value was easily seen to be $7$. Note that this was an informal calculation! The point is that without any further knowledge, you cannot tell what $\frac{1^{\infty}}{1^{\infty}}$ or $\frac{0}{0}$ is.
