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So in this problem we are given the function F
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. Of X equals two x cubed plus X.
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We were asked to determine if prime at a.
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Okay well by definition F. Prime at A.
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Is the limit As a church goes to zero of
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F. Of a plus H minus F. Of
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a. All over H. Okay so first of
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all we got to figure out half of a plus
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H. Well that's two times A. Plus H
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cubed plus A plus H. So that's two times
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E cubed was three A squared H. Plus three
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A. H squared plus one O. C.
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M. P. H. Sorry And that's plus
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A. Plus H. And so this becomes,
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well I have to a cute and I have three
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A squared H. And I have three A.
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That's not three. Sorry that's what babies and this
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one This will be six and that would be six
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. And then plus and a. Okay this is
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H cubed over here was and each plus an h
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cubed wouldn't it? Okay and let me just make
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this a little clear. Sorry about that. Just
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make this clear. That's too H cubed plus a
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. We'll see each. Okay F it a is
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simply to a cubed was hey isn't it? Okay
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so that means that F. At A plus h
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minus F. A. R. Numerator up here
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. Right our numerator up here. Well let's see
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The two A Cubes are going to cancel out.
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I'm going to have six A squared H. Plus
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six A. H plus two H cubed. Hey
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minus A. That will be gone. So I'll
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be plus H. Left on the end and when
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it, okay so that means now that I'm left
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with F. Prime at A. Is the limit
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as H. goes to zero Of six A.
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Squared H. Plus six h. plus two H
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. Cubed plus H. All over H. And
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now what do I see that age cancels? And
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that one cancels one of those cancels and that cancels
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doesn't it? Okay so yes I'm now left with
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this is the limit His age goes to zero of
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six A squared plus six A. Plus two eight
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squared. Well not mess up up here something didn't
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work out right I left two squared off of that
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turn right there didn't I? So that one squared
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. So this one was squared and so it should
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have been and H. Left there. Let's fix
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that up. This is the age squared cancel one
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of those out. Okay So this is six.
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Let's fix this up. Six A. Aah plus
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two H. Squared. Um So when I canceled
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this All right here that would have left or one
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. Okay so this is plus one down here.
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All right so now H. Goes to zero.
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So a return has an ancient it goes to zero
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. This one goes to zero and that one goes
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to zero So much with six A. Squared plus
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one is F. Prime today