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\question % workshop bank Ch. 4 -> esergel43

A form is indeterminate if it can take on more than one value. In parts (a)-(c) below, show that the given form is indeterminate. In other words, find two limits which have this form but evaluate to different values.

\begin{parts}

\part Show that the form $\frac{\infty}{\infty}$ is indeterminate.

\part Show that the form $0 \cdot \infty$ is indeterminate.

\part Show that the form $0^0$ is indeterminate.

\part Explain why the forms $\frac{0}{\infty}$ and $0^\infty$ are not indeterminate. Your answer should consider what happens when the parts of these forms are not precisely 0 or $\infty$, but very close.

\part We have seen in class that the form $1^\infty$ is indeterminate. For instance, in class we showed that $\lim_{x \to \infty} 1^\infty = 1$ and $\lim_{x \to 0} (1+x)^{1/x} = e$. However, a friend makes the following argument that $1^\infty$ is always $\infty$. What is wrong with this argument?
\vspace{6pt}

The form $1^\infty$ means we are taking a limit of $f(x)^{g(x)}$ where $f(x)$ is approaching $1$ and $g(x)$ is approaching $\infty$. But when you raise a number to a bigger and bigger power, it grows to infinity. For instance, $1.01$ is small, but $1.01^{1,000}$ is over 20,000 and $1.01^{10,000}$ is way over a billion. So $1^\infty = \infty$.

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