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In Exercises $55-62,$ use a method of your choice to find all horizontal and vertical asymptotes of the function.$$f(x)= rac{x^{2}+2}{x^{2}-1}$$...

Question

In Exercises $55-62,$ use a method of your choice to find all horizontal and vertical asymptotes of the function.$$f(x)= rac{x^{2}+2}{x^{2}-1}$$

In Exercises $55-62,$ use a method of your choice to find all horizontal and vertical asymptotes of the function. $$f(x)=\frac{x^{2}+2}{x^{2}-1}$$



Answers

In Exercises $55-62,$ use a method of your choice to find all horizontal and vertical asymptotes of the function.

$$f(x)=\frac{x^{2}+2}{x^{2}-1}$$

Okay. We know that the functions continues for all ex because X squared plus X is greater than equal to one for all values of X. Now we know that one over X squared approaches zero and three Other expert approaches zero Therefore, we have horizontal Aston Total the line. Why equals one? You have horizontal assam totes.

Here, we have the function f of X equals x times sine of X over X squared minus one. Uh The denominator you can never divide by zero. So you're going to run into problem affects this one because one squared minus one would be zero. And you're also going to run into trouble uh uh affects his negative one because if X negative one, negative one squared Because positive 1 -10 again, so X is not going to be able to equal negative one or positive one. So when we graph this As X Approaches one, Um you're going to end up having a vertical assam to at one and at -1. So at one we're going to have a vertical assume toe. The equation of this vertical red line is X equals one. And that's because as X approaches one uh this denominator is going to get closer and closer to zero. And uh this value up here divided by a number that gets really close to zero. When you divide by a small number, you get a very big number. For example, if I had a number like two And I wanted to divide it by .0001, you're going to get a very big number, something like 20,000. So as this number gets smaller uh this entire function gets very large, so it's going to go towards either positive infinity or negative infinity on this case is probably going to go towards positive infinity As we get close to one. Uh to function when we grab this function will use does most of them in to grab this function when we grab this function. Uh The graph of the function will never touch this red line. That's what makes it a vertical ascent. Okay? Um and ask them to is a line that the function gets closer and closer and closer to without ever actually touching. We found these tou assam topaz by looking for uh 22 values of X. That would make this denominator zero. So we're going to have to vertical aspect oaks. One at X equals 11 at X equals negative one. Now, let's take a look at the graph of this function in dez mose. Looking at the graph of this function in dez mose. Um You can see that X equals one is going to be a vertical assam. Topaz. X gets closer to one. Uh This curve shooting up towards positive Finney gets closer and closer to the line. X equals one. We can go ahead and plot that X equals one. So there is uh one vertical assam. Tope, the other one is at X equals negative one. So even though it looks like the graph touches this blue line and this green line, 22 vertical assam. Tokes X equals one. X equals negative one. It really doesn't. You can move up, scroll up and zoom in. Uh The graph will get closer and closer and closer to this line but never touch it. The graph will get closer and closer to this green vertical ass. In Tokyo X equals negative one. Uh Some toe but never touch it. Now you can see that the function gets close to zero. Um The line Y. Equals zero D. X. Axis. And actually if we zoom in uh it actually keeps uh you know, oscillating a little bit above it, a little below it. So it keeps touching uh this line and asked them toque is only a line is a line uh that the function gets closer and closer to but never touches. Uh This horizontal line. The X axis cannot be a horizontal assam Toby because the graph does touch it. A national top line is aligned that the graph gets closer and closer to but never touches. Ah So we only have uh to as some tops in this uh for this function and they both happen to be vertical assam tops.

Drenched you, Fracked and Andy Nominator I have one. And that front that x squared outside and inside. I have not thanks plus one and doesn't imply Stand doing, ma'am. He called you on your number. Except for the Baron is a rope and man a swarm. Understand? Means them. The bodyguard stood up. Randy noted by Feei will be Thanks, Coach Izzo on next Go to minus one Now for the heart that I should talk to play. I had to find a limit on a function Will execute your infinity number one another and tremors Think square Breda Wa s x g h infinity The This one goes to zero. There found a up A wait. Why would you zero?

So here we have a certain function. Half of x equals one over X -1 minus two X. So first of all, it's important that we write this in simplest form in order to determine a syncope sore point discontinuities. This will be one over X -1 -2 x Times X -1 Over X -1. So this is equivalent to one minus two X squared plus two X Over X -1. So let's first bring out the negative sign. So this would be two X squared minus two X minus one Over X -1. So let's first attempt to factor the top factor. So let's try to fast see if there's any factors and we see that there is a factor here. So this is important since if we have something in the numerator and something in the denominator. This is not essentially a direct ass into it. This is an example of a points continuity. And now, since we have nothing within the denominator, we don't directly have any vertical ascent coats in this case. Yes. And in regards to our horizontal ascent oats we have the following function in this case negative two X plus one X is just not equal to one. However, we see that there's not really a bounded limit on your why your ffx value, which means that we don't have a horizontal assam tote. Yeah, So we have no assam coats directly except for possibly an oblique ascent. Okay, And we only have point is constantly reduce and this gives our final answer


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