Analytic View - Tables

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\todo }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \DeclareMathOperator {\dx }{dx} \DeclareMathOperator {\dt }{dt} \DeclareMathOperator {\dy }{dy} \DeclareMathOperator {\dz }{dz} \DeclareMathOperator {\dr }{dr} \DeclareMathOperator {\dw }{dw} \DeclareMathOperator {\du }{du} \DeclareMathOperator {\dv }{dv} \DeclareMathOperator {\ds }{ds} \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\point }[1]{\left (#1\right )} \newcommand {\pt }[1]{\mathbf {#1}} \newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}} \newcommand {\bar }[0]{\overline } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

We discuss how to use tables to determine limits and the inherent danger to using this method.

Video Lecture

The basic idea

The natural starting point when thinking about limits analytically is to get actual values from the function “near” the limiting value, and to do this we use tables. Indeed, tables can be used to help guess limits, but one must be careful.

1 : Consider $f(x) = (1.01)^{\frac {1}{x}}$ . Fill in the tables below rounding to three decimal places: $\begin {array}{l|l} x & f(x) \\ \hline -0.1 & \begin {prompt}\answer {0.905}\end {prompt}\\ -0.01 & \begin {prompt}\answer {0.37}\end {prompt}\\ -0.001 & \begin {prompt}\answer {0}\end {prompt} \end {array}$ We may rush and say that, based on the table above, the limit of $f(x)$ as $x$ goes to $0$ is $0$ . But, recall our (intuitive) definition of the limit: $L$ is the limit of $f(x)$ at $a$ , if the value of $f(x)$ is as close as one wishes to $L$ for all $x$ sufficiently close, but not equal to, $a$ .

From this table we can see that $f(x)$ gets as close as one wishes to $L(=0)$ for some values $x$ that are sufficiently close to $a(=0)$ . But this does does not satisfy the definition of the limit, at least, not yet.

Indeed, let’s consider a couple more values (again, round to 3 decimal digits); $\begin {array}{l|l} x & f(x) \\ \hline 0.1 & \begin {prompt}\answer {1.105}\end {prompt}\\ 0.01 & \begin {prompt}\answer {2.705}\end {prompt}\\ 0.001 & \begin {prompt}\answer {20959.156}\end {prompt}\\ \end {array}$ What do these two tables tell us about the limit of $f(x)$ as $x$ goes to $0$ ?

The limit is $0$ The limit is $1$ The limit is $\infty$ The limit is $-\infty$ The limit does not exist.

The limit does not exist. The first table shows that we can always find a value of $x$ to the right of $0$ so that $f(x)$ gets as close as we want to $f(x)=0$ . However, the limit is not equal to $0$ , since the second table shows that we can also find a value of $x$ to the left of $0$ as close as we want to $0$ such that $f(x)$ is as big as we like (definitely not close to zero!).

As another example of how data can be misleading, recall the following graph:

When looking at a graph, it can be easy to see what is happening, because we have all the information except the one single point we want to know about. In the graph above it should be pretty clear what the value of $f(x)$ should be at $x=0$ , because we have all the values in the area around, including the values arbitrarily close to $x=0$ . But the solid line of the graph represents an infinitely large number of data points. Clearly there aren’t many (or any!) experiments where we get an infinitely large dataset. So in reality we are much more likely to get a graph that looks something like this:

The picture above is just a scattering of dots, which makes it difficult to get a sure sense of the pattern. “But surely it’s obvious that it is coming to a maximum at $x=0$ ” you might say, after all we’ve seen the original curve. But herein lies the problem; there are a lot of possible curves that could go through those dots! Consider the following graph:

This graph hits all the given data points, but clearly does something completely unexpected at the $x=0$ area. Why didn’t we detect this striking change in behavior? Because we didn’t have any data points whose $x$ values were “suitably close” zero! In fact, ensuring that you have data “close enough” to the $x$ -value of interest is actually almost always the hard part about limits!

In the video below we explore this issue of being “close enough” in more detail.

[YouTube video placeholder!]

2 : What is usually the difficult part about determining a limit in practice?

Figuring out which curve represents the function best. Knowing how close is “close enough” to get data points so you can determine the correct pattern. Algebra. Too much algebra. Graphing the data points precisely enough to know the value of the limit.

Optional Content

If you are interested in how mathematicians actually prove that values are “close enough” in general (not just for limits!) you can watch the following video on how mathematicians tackle the problem of being arbitrarily close rigorously. [URL to YouTube video, but not an embedded video.]