WEBVTT
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given the piecewise function F of x which is equal
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to two, X squared minus five X minus three
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over x minus three. If x is not equal
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to three and f of x equals six. If
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X equals three, we need to know why the
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function is not continuous at A equal to three to
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do this, we begin by checking conditions for continuity
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and the first one would be is F defined at
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x equals three. Now, since actually goes three
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is part of the domain of this function, then
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we have F of three. This is equal to
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six and this is defined. I actually want to
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verify if the limit of this function As X approaches
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three exists And since the limit as X approaches three
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of F of X, this is equal to The
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limit as X approaches three of two, x squared
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-5, X-3 Over X-3 which simplifies to
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The limit as x approaches three of two, X
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plus one times x minus three over x minus three
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, Which cancels out the X-3. And gives
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us the limit as X approaches three of two,
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X plus one and evaluating gives us two times three
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plus one equals seven. We say that the limit
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exists and unless you want to check if The limit
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of the function equals the value of the function at
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three. Now, since the limit of F of
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X As X approaches three equals 7, Which is
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not equal to six, which is the value of
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the function at three, then you say that this
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condition does not hold. And So the function is
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discontinuous at three. And for the graph of the
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function we will use the information we had from the
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previous steps. So in here we have f.
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F three, which is equal to six. The
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limit of the function as x approaches three equal 7
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. And previously have shown that f of X can
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be reduced to X plus one, Where X is
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not equal to three. So the first thing we
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have to do is to plot the point F of
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three equal six, which will be over here.
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Next we want a plot the function two, X
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plus one where X is not equal to three and
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we not hear that the function has a whole at
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X equals three. Because when we reduce this function
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, We were able to cancel out X-3.
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Now to find the y coordinate of this whole,
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we simply plug in X equals 3, 2.
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The reduced form of the function. So it will
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be what is going to be two times three plus
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one. That's equal to seven. So the whole
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has coordinates 3, 7. And we can put
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it here, which would be this point. And
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then to plot F of X which is equal to
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two, X plus one. We simply pick X
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values and we solve for F for the Y coordinate
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. So let's say X is equal to two and
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we have FF two, that's two times 2 plus
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one That's equal to five. And if XAT equals
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four, then we have fo for which is just
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two times four plus one, or that's nine.
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And so we plot 25 which will be this point
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, And then we plot 4 9, which will
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be this point. And then from here we connect
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the dots and we have the grab for two X
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plus one. And so this is the graph of
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our function.