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Current time:0:00Total duration:7:06

Analyzing vertical asymptotes of rational functions

Video transcript

whereas to describe the behavior of the function Q around its vertical asymptotes at x equals negative 3 and like always if you're familiar with this I encourage you to pause it and see if you can get some practice and if you're not well I'm about to do it with you all right so this is Q of X it's defined by a rational expression whenever I'm dealing with asymptotes I like to factor the numerators in the denominator so I can make more sense of things so the numerator here what two numbers well if I were the product is 2 and their sum is 3 well that's 2 and 1 so I can factor this is X plus 1 times X plus 2 if that's unfamiliar to you I I recommend you watch the videos on Khan Academy on factoring quadratics and over X plus 3 and X and when x equals negative 1 or x equals negative 2 would make the numerator equal 0 without making the denominator equals 0 so those are points where the function is equal to 0 but when x equals negative 3 the denominator equals 0 while the numerator is not equal to 0 so we're dividing by 0 and so that's a pretty good sign of a vertical asymptote that our function as we approach that value is either going to pop up like that or it's going to pop down or it's going to pop down like that or maybe or either way or it could pop up like that or it could go down something like that but we're going to have a vertical asymptote that the function as you approach x equals negative 3 that the function is going to approach either positive infinity or negative infinity or it might do positive infinity from one direction or negative infinity from another direction and if this idea of directionality is a little bit confusing well that's where we're about that's what we're about to address in this video so let's just let me just draw a number line here that focuses on these interesting values so we care about x equals negative 3 and then the other interesting value is maybe negative 2 negative 1 it's all there now what does it mean to be approaching X what is it what does it mean for X to be approaching negative 3 from the negative direction and just to be clear this little superscript right over here that means we're approaching from the negative direction so that means we're approaching from values more negative than negative 3 so those are these values right over here we are approaching we are approaching from that direction another way another way to think about is we approach from the negative direction we're on the interval X is less than negative 3 so let's think about what the what the what the sign of Q of X is going to be as we're approaching negative 3 from that negative direction from the left well this if we have something less than negative 3 and you add 1 this is going to be negative if you have something less than negative 3 and you add 2 that's going to be negative as well if you have something less than negative 3 and you add 3 well that's going to be negative as well so negative times a negative is a positive and then you divide by a negative it's going to be a negative so Q of X is going to be negative on that interval so as we approach and we have our vertical asymptotes we approach negative 3 from the left-hand side well Q of X is going to be negative and so it's going to approach negative infinity so as as X approaches negative 3 from the negative direction Q of X is going to approach negative infinity so at least this is accurate this is accurate and this is accurate right over here and you can validate that try some values out try try negative let's see if you did negative 3 point you know Q of Q of negative 3 point negative 3 point 1 once again that's on the left side it might be like right over there actually probably be a little bit too far it might be something like this like the scale the scale that I've drawn it on Q of negative 3 point 1 if you want to verify it it's going to be negative three point one plus one which is negative two point one times negative three point one plus two which is negative negative one point 11.1 I could put that in parenthesis just to make it clear and then all of that is going to be over well negative three point one plus three is going to be is going to be negative zero point one so notice whatever we get up here this positive value we're essentially going to if we're dividing it by negative zero point one that's like multiplying it by negative ten so it's going to get is going to become a very negative value and if instead of instead of it being negative three point one imagine if it was negative three point zero one then this would be a zero one here this would be a zero one here and this would be a zero one here and so this denominator you're dividing by negative zero point zero one is going to be an even larger negative value so you're going to approach negative infinity so it's going to be one of these two choices now let's think about what happens as we approach X from the positive direction and that's what that's what this notation over there means this superscript on the right hand side the positive direction so we're going to approach X from the positive direction and I'm going to pay attention in particular to the interval between negative two and negative three because then we know we don't have any weird sign changes going on in the numerator so I care about the interval negative 3 is less than X which is less than negative 2 so I can draw an open circle here say we're not we're not considering where when we're at negative 2 and of course we're not going to include negative 3 because our function isn't defined there but over this interval so X plus 1 is still going to be negative x plus 1 is still going to be negative if you took a negative 2 point 5 plus 1 it's going to be negative 1.5 X plus 2 is still going to be negative you're taking values that are more negative than negative 2 that are less than negative 2 so you add 2 to that you're still going to be negative and then let's see if you add 3 if you add 3 to these values remember they're greater than negative 3 or you could say they are less negative than negative 3 well then this is going to give you a positive value this is going to give you a positive value and think about it Q of Q of negative 2.99 what is that going to be equal to if you add 1 to that that is negative 1 point 9 9 and if you add 2 to that x times negative 0.99 all of that over negative 2 point 9 9 plus 3 well that's going to be 0.01 so you get a positive value on top and then you're going to divide it by 0.01 that's the same thing as multiplying by 100 so you're going to get larger and larger values you're going to approach infinity as you get closer and closer to it from the right-hand side so Q of X is going to approach positive infinity so this is the choice that is correct this one is wrong this says it's going approach negative 2 so that's incorrect and we will go with that choice