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DM
Numerade Educator

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Problem 2 Medium Difficulty

(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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Video Transcript

So in this question we were asked to crast if the function Y equals F of X intersects a vertical ascent. Okay. Yeah. And the answer is no. Hi. And why is that? Well, because Y equals F of X means for all X. In the real numbers there is a why value such that Y equals F of X. Okay, if we have a vertical assam tote then mhm. Then first of all, let's see. Well then why goes to infinity or minus infinity? At Uh huh. At that X. Okay. So there is no. So there is no value of why they're that occurs in other words, why does not intersect all along? Uh huh. X equals that value will call it a. Okay. Then. Were asked if can intersect the horizontal assam toe and the answer is intersect horizontal. Some toast. Yeah. And the answer here is yes. So let's think about a function here for a minute. We'll have our X. And or Y here. And let's say that our function does something like this like this, right? Where we have crossed this horizontal assam. Tote of Y equals zero, Husband crossed here and here and here. So it crossed at least three times with this function Because really the question is when X goes to infinity then Y equals F of X, right? Goes to why call zero is what happens? So you could you can't cross the horizontal assam toe at least once. Well, the function that is possible and still have a horizontal assam tote. It was demonstrated in that graph right there. So how many horizontal assume totes can a graph of Y equals F of X have? How many horizontal have some totes? Uh Some, hang on. MS builders correctly, awesome. Totes can we have? Well, think about it. We can have one as X goes to infinity, then Y equals F of X. Can go to horizontal. Have some tote. So there's one. What about when X goes to minus infinity? We could do the same thing Y equals F of X. Could go to another horizontal assam Tope. So there's a 2nd 1. So thus two horizontal have some totes are possible. Mm. Okay.