Speaker A
In this video, we're going to talk about the limit of a function as I approach a point for a multivariable function like f of xy equal to xy or x squared plus y squared. Now, if I look at this
Explore limits of multivariable functions using path restrictions and the two-path test to understand when limits do or do not exist.
Key Takeaways
- Limits in multivariable calculus can depend on the path taken to approach a point.
- The two-path test is a reliable method to show when a limit does not exist.
- If limits along all paths are the same, further analysis is needed to confirm the limit exists.
- Restricting multivariable functions to single-variable paths simplifies limit evaluation.
- Curved paths may affect limits differently than straight-line paths, posing further questions.
What the video covers
- Introduction to limits of multivariable functions and challenges at points where functions are undefined.
- Graphical interpretation of limits near problematic points, emphasizing approximation limitations.
- Comparison with one-dimensional piecewise functions to illustrate when limits exist or do not exist.
- Using path restrictions (e.g., setting y = x) to reduce multivariable limits to single-variable limits.
- Demonstration that different paths (e.g., y = x vs y = -x) can yield different limit values.
- Explanation of the two-path test to prove a limit does not exist if limits differ along two paths.
- Discussion of open questions about limits existing when all paths yield the same value.
- Consideration of whether curvy paths affect limits differently than straight-line paths.
- Encouragement to engage with the content through comments and likes.
- Mention of a larger playlist on multivariable calculus for further learning.
Chapters
- 00:00Introduction to multivariable limits
- 00:23Problematic point and limit analysis
- 00:47Graphical approximation of limits
- 01:13One-dimensional limit examples
- 01:30Returning to multivariable limit problem
- 02:09Restricting to path y = x
- 02:32Limit definition in higher dimensions
- 02:54Restricting to path y = -x and results
- 03:13Graphical and algebraic changes on different paths
- 03:41Multiple path limits and the two-path test
Full Transcript — Download SRT & Markdown
Speaker A
function, I notice that it is not defined at the point x equal to 0, y equal to 0. There becomes a 0 in the denominator at that particular point.
Speaker A
And as a result of this, this is going to take a little bit of analysis if I want to understand what does the limit as my xy gets closer and closer to the value of 0 0, what is the limit in that
Speaker A
case? If we look at this graphically, at the point 0 0, this is where we have the sort of weird pinching together of these sort of two different portions of the graph. And again, the question of what's going to be the limit as we get closer
Speaker A
there is going to stand. Indeed, when I try to graph something like this, it's only an approximation because the computer is interpolating between a bunch of points nearby but never equal to 0 0.
Speaker A
To help me understand this, let's go back to the one-dimensional case. Consider this piecewise-defined function. It's the value 1 when x is less than 1, and it's the value 2 when x is greater than or equal to 1.
Speaker A
Well, in this case, we say the limit does not exist. Indeed, the limit from the left when x is less than 1, that's easy to compute. That's just equal to the value of 1. But the limit from the
Speaker A
right when x is greater than 1, this is equal to 2. So, the limit from the left and the right do not match, so we say the limit did not exist.
Speaker A
However, in this different example, where the function is not defined at one point, but to the left and to the right, the function is indeed just equal to 1.
Speaker A
In this case, we said that limit did exist and was equal to the value of 1.
Speaker A
So, let's return to our multivariable case where we have this problem spot, that spot 0 0. We want to investigate the limit as I get close to that particular point.
Speaker A
One way I can start to grapple with this problem is to restrict myself to a particular direction, a particular line.
Speaker A
So, let me demand that Y is equal to X. This is a restriction and it's going to make this multivariable function just a single variable function just in terms of X that we know how to do. Indeed, everywhere there's a Y, we come and plug
Speaker A
the X value in and I'm just left with X squared over X squared plus X squared and then that's just X squared over 2 X squared which is 1/2.
Speaker A
Now, the real strength of what we did is that I did not have to give you a new definition for what the limit meant in a higher dimensional context. I just put a restriction on and then it became the
Speaker A
old one-dimensional concept of a limit and we computed this out. However, what if I change my restriction? I don't have Y equal to X, I impose, for example, Y equal to minus X. Well, watch what happens.
Speaker A
When I do this, there's a graphical change and an algebraic change. Algebraically, the equation Y equal to minus X just means I get a whole bunch of minus signs everywhere through my calculation and the result of that is
Speaker A
that I get the value of minus 1/2 for my limit. So, along this line, it's minus 1/2 whereas on the previous line, it was plus 1/2.
Speaker A
Now, graphically what's happened when I'm restricting myself Y equal to minus 1/2 is now that I'm on a new line, the heights along that line are quite a bit different. They're much lower down than they used to. That's why we have this
Speaker A
limit value of minus 1/2, but smaller than it used to be for plus 1/2. Indeed, if I focus just on that line, if I change that line from Y equal to X to a bunch of different lines, what we see is that the limit, the spot
Speaker A
in the middle where there's sort of a break in this particular function, there's this spot where it's undefined, that that approaches closer and closer to a bunch of different values depending on the line you restrict it to.
Speaker A
So, what this makes me think is that there is no limit for this function as X goes to the point 0 0. When you approach on some lines, it looks like it's getting close to a value, but if you
Speaker A
approach it in a different way, aka along a different line, then you get to a different value.
Speaker A
This is called the two-path test for a limit not existing. It says that if you go along one path, then you get some particular limit along that path.
Speaker A
And then you go along a different path and you get a different limit, then it says for sure the limit does not exist.
Speaker A
Now, that leaves me with two questions that we need to take up in the future.
Speaker A
Now, this test only tells us when a limit does not exist. So, the question is, if you go along one path and it is a value, and you go along a different path, and a different path, and a
Speaker A
different path, and they're always the same value, well, what exactly does a limit existing in the multivariable context mean?
Speaker A
That's the first question that we can think about for the future. And then the second question says that I've been focusing on straight lines, different straight line paths. And perhaps it was the case that every one of those straight line paths all gave
Speaker A
the exact same result. However, what if you approach it along a curvy path? Could that change the limit to something else, or does all that matter the straight lines? Again, a second question that I will leave you to think
Speaker A
about for the future. If you have a question about this video, leave it down in the comments below. We're all mathematicians here. We appreciate algorithms, so let's just help the YouTube algorithm out by giving this video a like. And finally, if you want
Speaker A
to watch more multivariable calculus videos, this video is part of a larger playlist on multivariable calculus, so you can check out those videos here, and we'll do some more math in the next video.
Topics:multivariable calculuslimitstwo-path testpath restrictionslimit does not existmultivariable functionscalculusmath educationDr. Trefor Bazettlimit along paths
