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(a) Can the graph of $y=f(x)$ intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.(b) How many horizontal asymptotes can the graph of $y=f(x)$ have? Sketch graphs to illustrate the possibilities.

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04:19

Daniel Jaimes

01:22

Carson Merrill

06:10

David Mccaslin

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Missouri State University

Baylor University

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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um Okay so we have kind of a interesting problem. We have two parts to the problem and the first part has to do with can can a function Y equals F. Of X intersect a vertical ascent toad. And so that's an interesting question. Um Usually the answer would be no. Um For example, if I write a function as let's say um one over X -1, then we get the graph that has a vertical ascent toad at one because they're not allowed to have the denominator go to zero. And it turns out that we get a graph, something like this. So notice we have a vertical as um toad and we don't cross the vertical as um toad. So a typical function would not allow you to cross. However, I could define a piecewise function that is for example one over X -1 for any location except at one because this is at X equals one where the ASM toad is and um go ahead and define say f. Of X zero at X equals one. So it wouldn't be exactly intersecting like crossing, but it would be physically on top of the vertical assam toast. So I would say in general no but if we're gonna count um actually being right on top, I can certainly define a point to go right there. But so that's kind of an interesting question. Um And the next part is really part of the same part is then kind of the similar question can Y equals F. Of X intersect um a horizontal a horizontal assam tow and this is definitely true. So um so for the first one I would say usually not. Um and for this one I would say yes it can happen. So for example, if we have, I pulled this one out, Just found a convenient one. If y equals three x squared -1 -2 over five x squared plus four x plus one. Then we get a just a Russia shape of the graph. Um As something that looks like with an innocent toad here At three fists. They have some toads at three fists. But then our graph looks something like this, it actually goes like above and then it goes down below and then it goes like this. So it definitely has the horizontal assume tote but we can cross it. Absolutely. So the second part. Absolutely yes. Alright, I'm going to clear the screen so we can do part B. So I will um Alright, so the second half to the question has to do with the maximum number of horizontal asuntos. And I'll answer clearly. It's too and the reason is is that to find a horizontal absent absent toad. You either check your function going to infinity and you can only get a single value or your check your function going to minus infinity. And uh a great example of a function that has two is Y equals or to make it Y equals ffx but it's inverse tan effect. This is a great function. Um If you ah if you graph it, you will find out that the assassin toads are the horizontal as some toads is. We have one at Positive pi over two and another at negative pi over two. And the function actually looks something like this. It's a pretty cool function goes like that. So um So there's an example of two horizontal sm totes uh in one function. So, anyway, hopefully that helped have an amazing day.

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