Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

Accepted Answer

Welcome back, everyone. Which are the following statements about the vertical and horizontal asymptotes of a function's graph is true. A that a functions graph can have any number of vertical asymptotes but no horizontal asymptotes B that it can have any number of vertical asymptotes. But at most two horizontal asymptotes C that it can have no vertical asymptotes and no horizontal asymptotes and D that it can have any number of vertical and horizontal asymptotes. Basically, what this question is asking us to do is to explain what we know about vertical asymptotes of a graph and horizontal asymptotes of a graph. Now let's make sure we understand both of these terms in order to figure out which of these statements is true. Let's start by looking at vertical asymptotes. What do we know? Well, if we do a quick sketch of our graph here, OK. X and Y as well, X and F FX. OK. Then a vertical Asymptote occurs where the value of a function F FX approaches infinity or negative infinity as X approaches a certain value. OK. Now, if we think about it here, this is what a vertical Asymptote would look like these dotted red lines. And a function can have multiple vertical asymptotes because it can approach infinity or negative infinity at multiple points. For example, let's take the function Y equals one divided by sine X. That function may look something like this. OK? That function is gonna look something like that. Let me put another sim to it here. And from this function, I mean, it's poorly drawn. But I think you get the idea from this function. You can tell that it has multiple vertical asymptotes based on how it looks. For example, in this portion of the function, this is its vertical Asymptote. While the other portion of the function on this side, it's the vertical Asymptote while the other is this one and so on. OK. So a function, let's just make a note of that here. A function can have multiple vertical symptoms. No. What about horizontal asymptotes? Well, when we talk about horizontal asymptotes, these occur as X approaches infinity. OK. Indicating the function's behavior at extreme values of X. So now if I were to again draw a line here or draw or a graph, OK, then for example, this line, this line could be considered a horizontal Asymptote. Now a function can approach a horizontal line from above or below as X approaches infinity or negative infinity. But it can approach at most one value from each direction. That means a function can have at most two horizontal asymptotes. For example, OK. If we take the function F FX equals the square root of X squared plus two divided by four X plus three. So a pretty seemingly complicated function, but if we break it down, that function is going to look something like this. Let me try this properly here. It may look something like this and then it may also look something like this. OK. So in this case, let me just clear this right here. Notice that it has two horizontal asymptotes. And if you were to actually plot the graph, then you may notice that as X approaches infinity F FX approaches a quarter. While if as X approaches negative infinity F FX approaches negative a quarter. But this is the most uh yeah, this is the most horizontal asymptotes that it can have. Therefore statement B is the correct choice as it accurately reflects the possible number of vertical and horizontal asymptotes. A function's graph can have. Thanks a lot for watching everyone. I hope this video helped.

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Determine whether the following statements are true and give an explanation or counterexample.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.