L'Hôpital's rule: limit at infinity example

we need to evaluate the limit as X approaches infinity of 4x squared minus 5x all of that over 1 minus 3x squared so infinity is kind of a strange number you can't just plug in infinity and see what happens but if you wanted to evaluate this limit what you might try to do is just evaluate if you want to find the limit as this numerator approaches infinity you put in really large numbers there you're going to see that it approaches infinity that the numerator approaches infinity as X approaches infinity and if you put really large numbers in the denominator you're going to see that that also well it not quite infinity 3x squared will approach infinity but we're subtracting it so if you subtract if you know some non infinite if you subtract infinity from some non infinite number this is going to be negative infinity so if you were to just kind of evaluate it at infinity the numerator you would get positive infinity the denominator you would get negative infinity you would get negative infinity it's all right like this negative infinity and that's one of the indeterminate forms that lapa tiles rule can be applied to and you'll probably say hey Sal why are we even using l'hopital's rule I know how to do this without lava tiles rule and you probably do or you should and we'll do that in a second but I just want to show you that a lot Patel's rule also works for this type of problems and I really just want to show you an example that had a had an infinity over negative or positive infinity indeterminate form but let's apply l'hopital's rule here so if this limit exists or if the door if the limit of their derivatives exists then this limit is going to be equal to the limit as X approaches infinity of the the numeron the derivative of the numerator so the derivative the numerator is the Rivet of 4 x squared is 8x minus 5 over driven the denominators well derivative one is zero derivative negative 3x squared is negative 6x and once again when you when you evaluate it at infinity the numerator is going to approach infinity and the denominator is approaching negative infinity negative six times infinity is negative infinity so this is negative infinity so let's apply l'hopital's rule again so if the limit of these guys derivatives exist or the rational function of the derivative of this guy directed divided by the derivative of that guy if that exists then this limit is going to be equal to the limit as X approaches infinity of arbitrarily arbitrary switch colors derivative of 8 X minus 5 is just 8 derivative of negative 6 X is negative 6 and this is just going to be and this is just a constant here so it doesn't matter what limit you're approaching this is just going to equal this value which is what if we put it in if we put it in lowest common form or simplified form it's 4 over negative 4 over 3 negative 4 over 3 so this limit exists this was an indeterminate form and the limit of this derivative over this drew this functions derivative over this functions derivative exists so this limit must also equal negative 4 over 3 and by that same argument that limit also must be equal to negative 4 over 3 and for those of you say hey we already knew how to do this we could we could just factor out an x squared you are absolutely right and I'll show you that right here just to show you that you know that's not the only you know l'hopital's rule isn't the only game in town and frankly for this type of problem i probably my first reaction probably wouldn't have been to use l'hopital's rule first you could have said that that first limit so the limit as X approaches infinity of 4x squared minus 5x over 1 minus 3x squared is equal to the limit as X approaches infinity let me draw a little line here to show you that this equal is that this is equal to that not to this thing over here this is equal to the limit as X approaches infinity let's factor out an x squared out of the numerator in the denominator so you have an x squared times 4 times 4 minus 5 over X right x squared times 5 over X is going to be 5x divided by let's factor out an x squared out of the numerator so x squared times 1 over x-squared minus 3 and then these X Squared's cancel out so this is going to be equal to the limit as X approaches infinity of 4 minus 5 over x over 1 over x squared minus 3 and what's this going to be equal to well as X approaches infinity 5 divided by infinity this term is going to be zero super duper infinitely large denominator this is going to be 0 that is going to approach zero and same argument this right here is going to approach zero so all you're left with is the 4 and the negative 3 so this is going to be equal to negative or 4 over a negative 3 or negative 4/3 so you didn't have to do use l'hopital's rule for this problem