The Concept of a Limit
Limits at a Real Number
Where do these things come up?
Mathematics: What does it mean for a function to be ‘undefined’ at a point? Can we hope to understand and classify the ways this can come about?
Physics/Economics: the predictions of a theory return “undefined” at some value. Do we need new physics to describe this? Or is this just a sign we should look closer at our mathematical model?
Computer science: a formula you derived throws an error: is this because of some small technicality you forgot to account for, or a fundamental problem / instability in your method?
For the remainder of this lesson we will focus on trying to understand qualitatively what kind of behavior is possible for functions, how we can fruitfully categorize it, and look for precise definitions to guide our future work.
Removable Singularities
Consider the following function
Such behavior is called a removable singularity as we can remove the problem by just defining a value for
A word of warning: such a situation might seem ridiculously artificial, but this kind of behavior pops up in very important ways all over mathematics! Indeed, the main subject of our course - derivatives - have this exact behavior embedded right into their definition.
It’s also good to be aware that not all removable singularities necessarily have a nice and clean single formula when the hole
has been patched. The singularity below is also removable - even though
Jump Discontinuities
A different kind of behavior which may occur when studying a function’s behavior right near a point outside its domain is a jump discontinuity. The reason for the name is self evident from an example graph, the function abruptly jumps from one value to another.
The important nature of such a jump continuity is that no matter what we assign
Essential Singularities
When a Function “Blows Up”
Finally, we could have a function that doesnt converge, jump, or oscillate when it reaches an endpoint of its domain, but
rather shoots off to infinity. A classic example of this is the curve
However, not all functions that grow in magnitude without bound seem to have a well-defined limits: consider
And finally, things can be even worse than this: its possible to write down functions which are undefined at
Definitions: Left and Right handed Limits
Behavior from above: Right-Hand Limits
Behavior from below: Left-Hand Limits
The Limit of a function
Given these two notions, we can begin to lay down some pretty rigorous definitions, precisely categorising the types of limiting behavior we observed above.
A function is continuous at
if firstly is defined at (so that makes sense), and secondly as we g approach , the outputs of get really close to its output at . Said in symbols, .A function has a removable discontinuity at
if is not continuous at , but we can re-assign to some new value, which then turns into a continuous function.A function has a jump discontinuity at
if there is no possible value of which would make continuous at , but both the right and left hand limits and exist. (And thus, necessairly take different values).A function has an essential singularity at
if at least one of the right and left side limits at does not even exist.A function limits to infinity at
if when approaches from either side, the values of grow larger and larger without bound. A function limits to negative infinity at if both approaches find to get more and more negative without bound.
Limits at Infinity
So far we have been concerned with the situation that a function
Where do these things come up?
Mathematics: Oftentimes a global understanding of a difficult problem in mathematics can be achieved studying separately things that happen ‘nearby’, and things that happen ‘far away’. Computationally this leads to a method of tackling problems by gluing together information from a small number of explicit computations, and two limits out to infinity.
The Sciences: In mathematical modeling, its often quite important to understand the long term behavior of a system. Left to their own devices, which species will eventually come to dominate this ecosystem? In the long run, what are the economic implications of a policy that’s being debated? Idealiziations of these questions involve limits as
(or , if instead we are trying to understanding how a system got to where it is now).Computer Science: To implement an algorithm on a computer, we need to first understand the behavior of the algorithm over the set of all possible inputs so that we can design our implementation accordingly (make sure we have the right data types, or memory, etc to deal with the whole range of potential results). Even in the simplest cases (algorithms where the inputs are numbers), understanding foundational metrics of computational complexity involve thinking about limits to infinity (in the size of input data).
Definitions and Examples:
Definition of
The function
Instead of having a finite limit at infinity, we may also encounter situations where
Finally, a function may tend to no limit whatsoever as
Functions can do this in a variety of ways, and need not stay bounded as they do so.
Here the function