00:01
In this problem, we are going to talk about the possibilities of intersection between the graph of a function and its asymptotes, either vertical or horizontal, and discuss about how many horizontal ascentopes can the graph a function have.
00:23
For that, we're going to do some sketching to prove or disprove some of these possibilities.
00:30
So the first one is, can the graph of a function intersect a vertical asymptote? in this case, it's very important to notice that we are not talking about functions defined by formulas.
00:45
There is no restriction about that.
00:47
We're talking about functions.
00:49
That's the only restriction.
00:52
It means that we can imagine graph with these properties.
00:56
For example, we can imagine a graph of a function that intersect a vertical asymptote.
01:02
I'm going to sketch that right now.
01:06
For example, i have the axis here, x and y.
01:16
And now i'm going to draw here a vertical dot line through, for example, the value one if you want.
01:27
And i'm going to draw a function with this line as vertical asymptote.
01:35
And in this case, it's going to be an asymptote by from the right and that it means that the limit from the right of at this point is plus infinity in that sense that vertical line through one is a vertical asymptote to this function and now the other part of the function i'm going to draw it like this let's say the function is increasing and it attain a value here and that's the key thing that is the function is decreasing and it has an image at the value one.
02:21
An image, it doesn't matter the value here.
02:24
You can take negative two if you want.
02:26
But the important thing is that it is intersecting the vertical line, the vertical asymptote at this point here.
02:36
I'm going to draw it in red to know what point i'm talking about.
02:42
So we have done a graph with vertical asymptote.
02:46
At the graph is intersecting at that vertical asymptote.
02:52
But in this case, as we can see, if we want to give formulas or expression, algebraic expression for this function, we got to do it piecewise.
03:05
That is, it cannot be maybe only one formula, but that's not the key thing.
03:10
The key thing is that this is a function, and this function has a property that its graph is intersecting, the vertical asymptote of the function.
03:21
So it can be.
03:23
The first question, the answer is yes here.
03:30
And this is the example.
03:35
For the other possibility intersection of horizontal asymptotes is easier.
03:41
It's even easier because i can draw only one piece of graph...