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In this video we#39;re going to go overl'hopital's rule and the different indeterminate forms you need to befamiliar with and we#39;re going to gothrough a ton of examples so you can master this topic so let's begin let'say if you want to find the limit as Xapproaches a for a function let#39;s say fof X divided by G of X and let'’s saythat this function is in the form 0 over0 or infinity over infinity these are known as indeterminate forms becausewhat exactly is 0 over 0 0 over 5 for example is 0 1 divided by 0 is undefinedso what exactly is 0 over 0 we done#39;tknow infinity over infinity could be constant like 4 it could be 1 it couldbe infinity it could be 0 we just done#39;tknow what the value is so they'rein determinate so let's say that thelimit as X approaches a for f of X is 0and for G of X is 0, so we would have theindeterminate form 0 over 0 if that#39;sthe case or if we have infinity overinfinity then this limit is equal to the limit as X approaches a F prime of Xdivided by G prime of X and that#39;s thebasic idea of l'hopital's rule if toucans#39;t find the limit in this form takethe derivative of the numerator and the denominator separately and you might beable to find the derivative in this form mean the limit in that form now thisis going to make sense once we do a few examples let's try an example what isthe limit as X approaches infinity forth function e to the X divided by xsquared well let#39;s see if we can usedirect substitution first so what is to the infinity e to a very large numberis going to be a large number it#39;sinfinityinfinity squared is also infinity so this is we what we have here is anindeterminate form which means that pecan use l'hopital's rule to figure this out because right now we don't know what that is now if you plug in numbers youcan also find out the answer let#39;s sayif we plug in a large number like 100 if you type this in your calculator if youcan do ite to the 100 divided by 100 squared is Avery large number it's two point sixeight eight times ten to the 39 direct substitution is like your fallbackmethod anytime you are having difficulty finding the answer this technique willalways work so if you plug this in this large number represents infinity soinfinity is the answer but let#39;s usel'hopital's rule to confirm of that answer so what we're going to do iswe'’re going to take the derivative ofthe numerator and the derivative of the denominator what is the derivative of eto the X the derivative of e to the X is simply e to the X and the derivative ofx squared is 2x now if we plug in infinity at this point it's still goingto be infinity divided by infinity two times infinity is infinity so we stillhave the indeterminate form so we cause l'hopital's rule a second timethe derivative of e to the X is e to the and the derivative of 2x is 2 sonotice that we no longer have the indeterminate form if we replace X withinfinity it#39;s going to be infinitydivided by 2 which is infinity and we know...
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