WEBVTT
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we want to find the limit of this expression as
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X approaches positive infinity. Now, when X gets
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very large on its way towards positive infinity, uh
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the numerator will be dominated by a particular term and
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so will the denominator. Um In the numerator we
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have two X squared plus one being raised to the
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second. When we look at the expression two x
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square plus one, the dominating term is going to
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be two X squared. And that's because as X
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gets very large towards positive infinity uh two X squared
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plus one will be dominated by two X squared effects
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is very large. X squared will be very large
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times in it by two will be very large adding
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one to that will not increase it that much.
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So the dominating term uh in this polynomial is two
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x squared. Uh So the top part of this
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function, the top part of this function is going
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to be dominated by the two x squared term.
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And keep in mind that that has to be raised
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to the second power. So we'll look at that
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in a moment. Yeah. Now in the denominator
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we have x minus one squared times X squared plus
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x in the x minus one. Uh expression,
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the dominating term clearly is the X as X gets
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very large X is going to get very large subtracting
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one from, it's not really going to change it
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. So the X-1 expression is dominated by the
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x term and then keep in mind we will have
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to square it. Now in this uh expression expert
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plus X. Uh the dominating term is going to
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be X squared because as X gets very large X
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squared gets even larger because she times in it by
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itself adding it to X does increase it because X
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is getting large but X squared is a whole lot
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more larger bigger than X because you're taking a large
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number and your times and by itself. So the
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dominating a term in this expression is two X squared
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. Uh So the numerator is being dominated by two
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X squared and there has to be raised to the
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second. Uh The denominator is being dominated in this
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expression by X, which has to be raised to
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the second and that's time zing. Uh The dominating
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term in this expression which is X squared. So
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to make a long story short, the numerator,
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this entire expression two X squared plus one to the
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second will be dominated by two X square to the
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second. The numerator is going to behave largely as
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the two X square to the second behaves now,
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two X squared to the second is really to to
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the second which is four times X squared to the
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second which is X to the fourth. So if
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X is moving towards positive infinity, the numerator is
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going to be behaving like four Times X to the
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4th. The denominator is going to be behaving as
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X squared times X squared does X squared times X
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squared is exited 1/4. So the denominator of this
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entire expression will behave as X to the fourth does
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uh as X approaches positive infinity now, four times
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X to the fourth, divided by X to the
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fourth. Those exit a force will cancel And that
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leaves you with four. And so the limit of
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this entire expression as X approaches positive infinity is equal
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to four. This entire expression will approach the value
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of four as X approaches positive infinity.