 Let's take a look at one more example of finding asymptotes to a graph and so maybe I have the rational function y equals 2x squared minus 5x plus 2 over x squared plus x minus 6. And so again as usual because it's a rational function we want to first find the forbidden values those things that are going to make our denominator equal to 0. So I'll solve that equation denominator equal to 0 x squared plus x minus 6 equal to 0 and this is a quadratic equation so I can use the quadratic formula and solve x equals 2x equals negative 3. So these make the denominator equal to 0 so we want to make sure that we indicate that these are forbidden values we are not allowed to let x be 2 or negative 3. After we've done that we want to divide so we can get a proper rational function so here I have the degree of the numerator is degree 2 and denominator also degree 2 so I do need to divide and importantly our denominator x squared plus x minus 6 is not of the form x minus a so I can't use synthetic division on it unfortunately and we have to use some form of long division and so we apply our long division algorithm whichever one we choose to use and we end up with this quotient of 2 and a remainder minus 7x plus 14 and we'll use the fact that we've already figured out that 2 and negative 3 are the roots of the denominator so I know that this denominator factors as x minus 2 times x plus 3. Now it turns out there's a little bit more simplification we could make here this minus 7x plus 14 well that turns out to be minus 7 times x minus 2 so I can actually simplify this this remainder even further and so my original rational function again keeping the prohibition x cannot be 2 or negative 3 this thing simplifies to 2 plus negative 7 over x plus 3 so there's my rational function in a somewhat more useful form so how about those asymptotes so again we care about two things the end behavior as x goes to plus or minus very large number and also what happens near the forbidden values x equals 2 x equals negative 3 so let's see as x goes to positive large as x gets very large very positive then y well let's see so that x is a large positive number so this fraction here is going to go to a small number so y gets close to 2 and importantly because minus 7 over x plus 3 as x gets positive large this is going to be a small negative number so y is going to be a little bit less than 2 so I might say that as x goes to plus infinity y approaches 2 but stays a little bit below it likewise as x goes to minus infinity as x gets large but negative again this rational expression here is going to go to a small number and so y is going to get close to 2 and as x gets large negative this term is going to be a small positive number so y is going to be 2 plus a small positive number y is going to be slightly more than 2 so I have my end behavior and the next thing I want to find out is x gets close to 2 I want to see what happens as I get close to the forbidden values y gets close to well nothing really bad happens here if x is close to 2 this expression minus 7 over 5 well that's just a rational number y gets close to 2 plus negative 7 fifths and importantly that is not plus or minus infinity and remember we only have an asymptote if either x is approaching plus or minus infinity or y is approaching plus or minus infinity so what that says is that x equal to 2 is not going to be an asymptote because nothing approaches infinity on the other hand let's take a look at that other forbidden value x equals negative 3 so as x gets close to negative 3 from a little bit below this expression here negative 7 over x plus 3 becomes a large positive number so y gets close to a large positive number y goes to plus infinity and as x gets close to negative 3 this time staying a little bit more than negative 3 then this becomes a large negative number so y is going to go to minus infinity and so our asymptotes are going to be y approaches 2 y equals 2 is going to be a horizontal asymptote and x equals negative 3 is going to be a vertical asymptote