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Investigating limits with Julia¶

To get started, we load the MTH229 package so that we can make plots and use some symbolic math:

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Quick background¶

Read about this material here: Investigating limits with Julia.

The expression

$$\lim_{x \rightarrow c} f(x) = L$$

says that the limit as $x$ goes to $c$ of $f$ is $L$.

Intuitively, as $x$ gets “close” to $c$, $f(x)$ should be close to $L$.

If $f(x)$ is continuous at $x=c$, then $L=f(c)$. This is almost always the case for a randomly chosen $c$ - but almost never the case for a textbook choice of $c$. Invariably with text book examples—though not always—we will have f(c) = NaN indicating the function is indeterminate at c. For such cases we need to do more work to identify if any such $L$ exists and when it does, what its value is.

In this project, we investigate limits three ways: graphically, with a table of numbers, and analytically, developing the inituition of limits along the way.

Graphical approach¶

The graphical approach is to plot the expression near $c$ and look visually what $f(x)$ goes to as $x$ gets close to $c$.

For example, what is this limit?

$$\lim_{x \rightarrow \pi/2} \frac{1 - \sin(x)}{(\pi/2 - x)^2}?$$

Here is a graph to investigate the problem. We simply graph near $c$ and look:

From the graph, we see clearly that as $x$ is close to $c=\pi/2$, $f(x)$ is close to $1/2$. (The fact that f(pi/2) = NaN will either not come up, as pi/2 is not among the points sampled or the NaN values will not be plotted.)

Using tables to investigate limits¶

Investigating a limit numerically requires us to operationalize the idea of $x$ getting close to $c$ and $f(x)$ getting close to $L$. Here we do this manually:

From this we see a right limit at 0 appears to be $1$.

We can put the above into a column, by wrapping things in square brackets (forming a vector):

The style of printing makes it clear, the limit here should be $L=1$.

Limits when $c\neq 0$ are similar, but require points getting close to $c$. For example,

$$\lim_{x \rightarrow \pi/2} \frac{1 - \sin(x)}{(\pi/2 - x)^2}$$

has a limit of $1/2$. We can investigate with:

Wait, is the limit $1/2$ or $0$? At first $1/2$ seems like the answer, but the last number is $0$.

Here we see a limitation of tables - when numbers get too small, that fact that they are represented in floating point becomes important. In this case, for numbers too close to $\pi/2$ the value on the computer for sin(x) is just 1 and not a number near 1. Hence the denominator becomes $0$, and so then the expression. (Near $1$, the floating point values are about $10^{-16}$ apart, so when two numbers are within $10^{-16}$ of each other, they can be rounded to the same number.) So watch out when seeing what the values of $f(x)$ get close to. Here it is clear that the limit is heading towards $0.5$ until we get too close.

For convenience, the lim function from the MTH229 package can make the above computations easier to do. Its use follows the common pattern: action(function, arguments...). For example, the limit of $1$ is clearly suggested below:

The limit of $1/2$ is suggested by the following

The above will generate values just bigger than pi/2 and just smaller than pi/2 which are helpful to investigate the limit from both sides. For just a left limit, pass in dir="-", as with

Similarly, pass in dir="+" to get just the right limit.

Symbolic limits¶

The add-on package SymPy can be used to analytically compute the limit of a symbolic expression. The package is loaded when MTH229 is. SymPy provides the limit function. A sample usage is shown below:

The new command, @syms x, creates a symbolic variable named x. This makes f(x) a symbolic expression

To find the limit of an expression, ex, as a symbolic variable, x, goes towards some value c, 0 in the above example, is performed by the command limit(ex, x => c).

SymPy finds the right limit by default. A left limit can be asked for with the additional argument dir="-". Passing dir="+-" to SymPy asks for both the left and right limit and a test for their equality (e.g., the limit).

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