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Vertical Asymptotes

Vertical asymptotes are straight lines of the equation , toward which a function f(x) approaches infinitesimally closely, but never reaches the line, as f(x) increases without bound. For these values of x, the function is either unbounded or is undefined. For example, the function has a vertical asymptote at , because the function is undefined there. Vertical asymptotes occur when the limit of a function approaches ±∞, known as a singularity. A function can have any number of vertical asymptotes, such as the function , which has them every radians, where n is an integer. Vertical asymptotes can be determined using one-sided limits.


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Lesson on Vertical Asymptotes

Vertical Asymptotes
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Calculus
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hey guys this is mohammed p and to their own Chegg.com today i’m talking about vertical asymptotes okay so where to go asymptotes are basically the vertical ones that when the X approaches twit like a specific value then at that specific value the Y is going easier to positive infinity or to negative infinity and it usually happens when you are when you are dealing with the rational functions and when the denominator becomes 0 okay so let’s look at a problem here that I have a y equal one over x squared minus 1 okay in order to find the vertical asymptotes of the rational functions what we are going to do we are going to set the denominator equal to zero so I’m sitting x squared minus one equal to 0 and then I’m trying to solve for x so x squared is going to be one and then X is either negative 1 or X equal one okay so these two results are going to be my vertical asymptotes and what do they graphically mean they mean that at x equal 1 and negative 1 so this is X equal negative 1 I would always withdraw the vertical asymptotes with a dashed line and X equal one so at these two x-values the function is either going to positive infinity or negative infinity let’s try some values I have actually two sides to approach here and two sides to approach here so i’m going to try like x going to a one from the right side x going one from the left side x going to negative 1 from the right side and x going to negative one from the left side okay for this case if i plug in a number a little bit bigger than one let’s say one point zero zero 1 so 1 point 0 0 1 squared plus 1 I mean minus 1 okay so in this case if i evaluate the denominator i would get 0 positive 0 positive it’s like it is very small number and close to 0 which is positive also and 1 divided I by ready a small number and positive it’s going to be positive infinity so this means as I’m approaching to the X equal 1 from the right side the function is approaching to positive infinity like this now if i plug in a number a little bit is smaller than one let’s say 1 over point 9 9 9 squared minus 1 if i calculate this i would get 0 negative 0 negative means it’s a number very close to zero but negative like a little bit less than 0 so 1 divided by a very very small number and negative it gives me negative infinity so it means that the function is approaching 2 1 from the left side and going down to negative infinity so as you see i have a vertical asymptote at x equal one from one side is going to positive infinity and from the other side is going to negative infinity so when we are finding the vertical asymptotes we are going to find the x coordinates of the line of the vertical lines that the function either goes up or down very close to that line so you can do the same thing for negative 1 from the right and left and you would get some result either positive infinity or negative infinity but I want to actually mention something very very important that’s really hard to see some people they don’t really know that’s a que I want to show you for the time that you get a zero when you set the denominator equal to zero but that’s not a vertical asymptote okay so let’s look at the example y equal x minus one over x squared minus one so very similar to the previous one but i’m adding x minus one on the top let’s see what’s happening so if i factor out the denominator I would get X minus 1 times X plus 1 obviously so in this case i can simplify x minus one and i would end up with one over X plus 1 so this is the thing that you need to always do you have to simplify the fraction first before setting the denominator equal to 0 because some factors they cancel out if you simplify and those factors are not going to give you the vertical asymptote anymore for example if I say x squared minus 1 equals 0 x squared is equal 1 then X equal plus 1 and X equal minus 1 huh that’s not the case because X minus 1 cancel and if you graph this function you are going to see I’m sorry here I need to say X plus 1 is not a vertical asymptote because here yeah we cancelled x minus one so if we graph this function we are going to see that at x equal one we are going to have a whole so the function is not going to have a vertical asymptote at this point because it’s already cancelled and there is no vertical asymptote anymore so whatever is cancelled from top and bottom of the function it is going to turn to a hole so it’s turning basically to a whole rather like a vertical dashed line which is vertical asymptote and the other one X equal negative 1 is still going to be a vertical asymptote because here and in whatever is left if i plug in negative 1 i would get 1 over 0 so it’s either plus infinity or negative infinity depending on the side you can check that ok so I hope that you are enjoy and you learned these important things that is really hidden from the eyes and I hope that you learn more things from my future videos I try my best to actually go over more topics and i will say goodbye till next time thank you

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