Indeterminate Forms
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►PRODUCT RECOMMENDATIONS https://www.amazon.com/shop/brithemat... (affiliate links) Indeterminate Forms are essential to solving L'Hospital's Rule. After you know these different indeterminate forms you will be able to solve almost any limit problem. Here are the indeterminate forms you should know.
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okay in this video we're talking about
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the seven different types of
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indeterminate forms are going to deal
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with in calculus and when using
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l'hopital's rule so I've broken them up
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into three different categories and this
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first category is the one that comes up
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the most and it's the one that's easiest
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to deal with
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because when you get zero over zero or
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infinity over infinity immediately you
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can just do l'hopital's rule okay if you
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haven't seen l'hopital's rule go ahead
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and check out my video on it should be
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somewhere really close so for example if
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you had the limit as X goes to zero of x
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squared minus x over I don't know
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X cubed plus X I don't know that would
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be a case where you're going to get zero
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over zero and you can immediately jump
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to l'hopital's rule or you could have
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something where limits going to infinity
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and I mean I could actually just use the
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same example here x squared minus x over
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X cubed plus X and then you would get
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infinity over infinity and immediately
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you would just use an apply l'hopital's
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rule so zero over zero or infinity over
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infinity those are the easiest cases
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because you can just immediately use
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l'hopital's rule the second category is
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just a little more difficult and
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basically you're gonna transform 0 times
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infinity or infinity minus infinity into
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one of these okay so your your goal is
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tutor to transform the second category
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into the first category so let me show
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you an example of something that would
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look like that so let's just say you had
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X going to 0 from the right of x times
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Ln X if I plug in 0 that would be 0 and
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then log of 0 or the natural log
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as we approach 0 from the right actually
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goes down to minus infinity so you have
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0 times minus infinity and the question
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is well is zero going as X going to 0
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faster or is Ln X going to minus
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infinity faster so that's why it's
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indeterminate we don't quite know but
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the strategy is you write this as a
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fraction ok so you've got this function
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times this function and all you do is
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instead of writing x the first function
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you write 1 over that function and these
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are completely equivalent statements but
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now the thing is if I plug in 0 I'm
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getting negative infinity and now 1 over
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0 is infinity I'm back at stage 1 I can
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just do l'hopital's rule so whenever you
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see 0 times infinity you write it as a
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fraction by writing 1 over one of the
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functions the other thing that can
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happen and it's a little less common is
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infinity minus infinity let's say you
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add the limit as X goes to 0 of 1 over X
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minus 1 over sine X and if I evaluate
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this thing at 0 I'm gonna get 1 over 0
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minus 1 over 0 1 over 0 approaches
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infinity 1 over 0 approaches infinity
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and the question is is this infinity
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bigger is this infinity bigger so that's
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why it's indeterminate and so your
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strategy for this is to combine these
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two things somehow you want to make it a
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single fraction instead of two different
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fractions so you could just combine
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these two fractions by saying cross
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multiplying 1 times sine is sine minus
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sign in the middle x times 1 is X and
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then you multiply the denominators
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together
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voila you've combined those two
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fractions and now if I plug in zero sine
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of zero is zero minus zero is zero and
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I'll get a zero on the bottom and hey
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look at that I'm in stage 1 again and
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then I can immediately jump l'hopital's
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rule okay
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all of these are just transforming back
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to step one and again if I'm going too
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fast please just give me a pause rewind
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watch the view again stage three this is
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the hardest this is when you have zero
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to the zero or one to the infinity or
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infinity to the zero so here's an
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example of something when that would
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happen if you had limit as X goes to 0
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from the right of X to the X actually
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made a video about this very problem
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it's a really common question asked so I
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have a video on this problem I'm not
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going to go through it right here but if
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you want to see this problem done you
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can check it out I've got it on my
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channel if I evaluate this I plug in 0
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for X I have 0 to the 0 that's an
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indeterminate form so what we do with
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all of these is we do this trick where
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we do e to the Ln I'm taking advantage
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of the property of logarithms that said
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if I have e to the log I can cancel
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those and just get what's on the inside
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well if I can go from right to left I
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can go from left to right so I haven't
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done anything wrong here it's just I
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kind of like x 1 in a sense so that's
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why I'm allowed to do this but now I
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take advantage of another property of
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logarithms that says if I have the log
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of something to a power I can write it
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as the coefficient it's not the power
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rule it's just a property of logarithms
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now if I plug in zero from the right
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I'll get e to the 0 remember we just
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said log of zero is negative infinity
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hey look what happened
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we got moved into category two we got
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zero times infinity what did we do when
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this happened well we wrote it as a
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fraction to make it into stage one so
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what you do from here is you write this
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like e to the log of 1 over X and now if
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I evaluate at 0 I get log of negative
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infinity over 1 over 0 which is infinity
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you're in stage 1 just like last time
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and then you'll use lhr just in the
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exponent of e so all of your answers are
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going to be e to something in this third
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category 1 to the 0 or 1 sorry 1 to the
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infinity and 0 to infinity they don't
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come up as much but they're the same
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thing you're gonna do this e & Ln trick
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and then bring down the power you'll be
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in stage 2 and then write it like a
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fraction and you'll be in stage 1 those
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are the seven different indeterminate
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forms I don't personally know if there
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are any others that you're gonna see in
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calculus but those are the most common
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for sure if you can do these you can do
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just about any limit problem in the
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world so there you go I hope you got
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something out of it if you want to see
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more indeterminate form videos please
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let me know I'm always wondering what
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you guys want to see me do and please
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like comment and subscribe I put out new
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videos just about every single day okay
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have a great day and thanks for watching