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welcome to another video Let’s Take a
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limit of x factorial over x to the X as
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X goes to
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Infinity obviously everything is getting
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bigger the top is getting bigger the
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bottom is getting bigger everything is
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driving toward Infinity at a supersonic
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speed and when we get to Infinity what
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are we going to get we’re going to get
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infinity
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over infinity which is an indeterminate
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form which you cannot do anything with
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unless you can do some factoring or you
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can do some litol rule unfortunately I
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don’t know any algebra to help me Factor
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as many X’s so that at the end of the
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day I have a simple expression to
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evaluate so what should I do well I am
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going to try lal’s rule unfortunately
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also I know how to differentiate the
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denominator but when I differentiate a
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factorial function I get some kind of
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weird derivatives which I don’t know
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what to do with because it introduces
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another weird function we even have to
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start from the gamma function in the
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first place
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so what can I do is there something else
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you can do yes there is it is called The
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Squeeze theorem however you may not know
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how to use the squeeze theorem properly
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if you don’t know what factorial means
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because that is where the problem is
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anytime you’re going to Infinity the
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denominator is usually not the problem
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the numerator is usually the problem
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let’s get into the
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[Music]
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video Let’s understand what x factorial
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actually means you see this is a
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function that says that if I told you
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what is 3
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factorial I would Define 3 factorial as
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3 * 3 – 1 which is 2 * 2 – 1 which is 1
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So my answer here is
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six let’s take a smaller number what if
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I said two factorial well two factorial
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is start from two you multiply by 2 Min
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1 which is going to be 1 so as you can
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see it is just you multiplying starting
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from that number you keep reducing it by
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1 by one by one by
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one we can go higher what if I wanted to
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know what one factorial
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is that’s hard to write 1 factorial is
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just
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one what about 0 factorial 0 factorial
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is just
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one so what about minus1 factorial well-
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1 factorial is not defined because see
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the factorial function is defined only
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for non negative integers so I’m talking
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of 0 1 2 3 4 you just keep going like
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that from zero to to to in to Infinity
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just keep going that’s what you define
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factorial for so it is certain that the
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smallest value you can obtain when you
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take a factorial is one and the biggest
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value you can obtain when you take a
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factorial is in inity so any factorial
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that you take is between 1 and infinity
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it is important for our squeeze
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theorem so watch this it means if I
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write X
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factorial what I’m saying is x * x
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-1 multiplied x – 2 and I keep
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multiplying and the last number I’m
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going to get is
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one
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now it is important to see what I’m
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about to show you whenever you do a
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factorial the number of terms that are
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really
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significant is one less than the actual
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number look at this 3 factorial is 3 * 2
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the one doesn’t change anything so you
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might as well say it is just 3 *
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2 okay what do you think happens if I
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refuse to subtract one from the next
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term I keep it as 3 *
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3 let’s let’s remove this since it’s not
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important see this is what I’m going to
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have this is going to be less than or
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equal to 3 * 3 6 is less than or equal
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to 9 it satisfies the less than option
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the same thing anytime you write
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something like this if I say you know
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what I don’t care about reducing the
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numbers I just want to keep it the way
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it is I know that that X
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factorial is supposed to be this it’s
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going to be less than if I write X and
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instead of writing x – one again I just
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write X and then I write X and then I
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write X and I keep going until I get to
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one what I have just written is X
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multiplied by itself how many
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times see remember there this minus one
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there are three terms I started from
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three but I only needed two terms to
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express my answer because the last term
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is one and doesn’t change anything so
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when I write this I’m going to write X
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ra to^
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x minus 1 I I wouldn’t do all of them
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see I didn’t do all three I only did two
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I didn’t do all two I only did one so
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whenever you write an exponent sorry um
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a factorial function it always has so n
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factorial is the same thing as has um
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let’s spread this way has nus one
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terms that you multiply together because
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the last term will be one which doesn’t
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affect anything so this is
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significant
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okay now that’s what I need here I’m
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going to make a
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claim I’m going to say
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notice that one is less than or equal to
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any factorial that you compute do you
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agree yes we just explained it the
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smallest factorial is
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one right and any factorial that you
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compute will always be less than or
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equal to this case it will be less than
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the case where you refuse to subtract or
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make it smaller you just keep it like
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that come on you’re supposed to reduce
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each of the subsequent terms but if you
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decide decide to not reduce each of the
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subsequent terms this is what you’re
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going to get but we know this is
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supposed to be this but if you replace
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it with this this has to be less than x
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to the x -1 and this is the key to
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taking this limit so we’re going to
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build the squeeze theorem from this
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since we have two inequalities now what
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do we do this is our Focus what do we
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need to make this look like this we just
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need to divide it by x to the X so we’re
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going to divide every term here by x to
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the X so we have 1 over x to the
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x is less than or equal to X factorial
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over x to the x and x to the X will
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always be positive because remember all
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the numbers we’re using here are
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positive okay positive now this is going
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to be less than or equal to X to x -1 or
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x to the X
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nice is there a way we can rewrite this
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this can be written as how do we
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simplify this is there
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simplification
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um we can actually divide we can take
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limit okay let’s just take the limit of
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this function as X goes to Infinity what
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we have is going to be the limit as X
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goes to Infinity
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of 1 /x to X is less than or equal to
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the Limit as X goes to
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Infinity of X factorial over x to the X
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which is our Focus Which is less than or
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equal to the Limit as X goes to Infinity
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of x to the X now watch this what is X
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to the
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x- one is the same thing as x to the X
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time um 1
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/x because this is X Theus one / x to
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the X so I just broke up what was on top
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and made it look like this and we can do
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some canceling well let’s simplify one
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more time this x to the X will cancel
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this x to the
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X and that’s it now we have the limit as
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X goes to Z Infinity of 1
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X is less than or equal to the
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Limit as X goes to Infinity of X
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factorial over x to the X and it’s less
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than or equal to the Limit as X goes to
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Infinity of
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1/x I’m sure you know what this limit is
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this is zero is less than or equal to
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what we have here and this is less than
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or equal to zero
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okay as X goes to Infinity of X
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factorial over x to the X
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nice so by The Squeeze theorem our limit
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is zero because both sides of this
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inequality are zero so we
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say therefore the limit as X goes to
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Infinity of X factorial over x to the x
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is equal to 0 by
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The
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Squeeze
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theorem never stop learning those who
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stop learning stop living
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bye-bye