This is the fifth project option for Calculus I during Fall 2017 at Fitchburg State.
This project involves ordering types of functions by investigating their limits at infinity.
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{\scshape Math 2300} \hfill {\scshape \Large Project \#5} \hfill {\scshape Fall 2017}
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In this Investigative Project, you will use l'H\^opital's Rule to develop a hierarchy of functions. Hopefully you've carefully thought about the questions and ideas posed at the end of the in-class worksheet on l'H\^opital's Rule. In the beginning, you compared the functions $f(x) = x^2$ and $g(x) = 2^x$. Include a careful write-up of this analysis here.
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You should have discovered that $\ds{\lim_{x \rightarrow \infty} \frac{2^x}{x^2} \rightarrow \infty}$. This means that the function $2^x$ is much larger ``in the long run" than the function $x^2$. We can convey this in symbols as $$x^2 \lessarrow 2^x.$$
In this case, the exponential function ``wins" this battle, but is that always the case? What happens if we have a larger power for $x$ in the power function and/or a smaller base in the exponential function? Is $x^{100} \lessarrow 1.5^x$? Try comparing a few different pairs of one power function and one exponential function. How low can you make the base of the exponential and/or how high can you make the power while maintaining the same $\lessarrow$ ordering? Determine a theorem, given it a name, and provide (if not a proof) a strong argument for your theorem. It might have a form that looks like:
\begin{quote} {\bf The \underline{\hspace{1.5in}} Theorem:} For any value $p$ where \underline{\hspace{.25in}} $ < p < $\underline{\hspace{.25in}} and/or any value of $b$ where \underline{\hspace{.25in}} $ < b < $\underline{\hspace{.25in}}, $$\lim_{x \rightarrow \infty} \frac{b^x}{x^p} \rightarrow \infty,$$
so $x^p \lessarrow b^x$.\end{quote}
and you will reference l'H\^opital's Rule in the justification. Note that your conditions on $p$ and $b$ maybe in a different form (like $b>4$) or there may be no restrictions on one or the other (for any real numbers $p$ and $b$ \dots). It is your job to try to push these bounds as far as you can.
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The activity outlined above compared the end behavior of exponential functions with the end behavior of power functions. Another class of functions that grow without bound as $x$ approaches infinity are the log functions: $f(x) = \log_a(x)$. Compare this class of functions with power functions and with exponential functions? Which type of functions ``wins" in the limit? Is this always true? Change some of the parameters like you did before and see what happens. Write and justify a theorem (or two) that summarizes your findings.
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Also, consider changes in the parameters within these classes of functions. Of course, if we increase the power just a tiny bit, that makes the function bigger, but how much bigger? Enough that $x^4\lessarrow x^{4.000001}$? How do these changes work in each class of functions? How much larger must $p$ be than $q$ so that $x^q \lessarrow x^p$? What about exponentials and logs?
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List an ordering of sorts of log functions, power functions, and exponential functions with a variety of powers/bases that demonstrates all the work you've done.
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{\small Note: I just made up the symbol $\lessarrow$, you can use a different symbol just be sure to define it clearly. }
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