Re: proving limit for given values of epsilon

In article
<0a32c94a-ffee-474d-a515-4434b94191a4@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
conrad <conrad@xxxxxxxxxx> wrote:

On Jan 27, 12:31 pm, The World Wide Wade <aderamey.a...@xxxxxxxxxxx>
wrote:
In article
<05b4cb61-e2c0-4d98-acb2-150b719ae...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,





 conrad <con...@xxxxxxxxxx> wrote:
For the limit

lim_x->1 (4 + x - 3x^3) = 2

I'm trying to find values of delta that
correspond to the epsilon values
1 and .1

I start off by finding a corresponding
delta for epsilon = 1
and expressing it this way:

|(4 + x - 3x^3) - 2| < 1
whenever
0 < |x - 1| < delta

Now I can approach the solution
to this problem by first making an
educated guess for what delta
would be. In this case,
I can started by factoring
2 + x - 3x^3

This is where I am stuck though.
This part seems convoluted.
I know one factor of this polynomial
is (x - 1) and the other two are imaginary
factors (3 +/- sqrt(9 - 24))/-6

Am I going about this all wrong?

--
conrad

All of these polynomial problems are the same. You want to show
lim_x->a p(x) = p(a). We have p(x) - p(a) = (x-a)q(x) for some
polynomial q. We next cleverly observe that |q| will be bounded by
some constant C in the interval (a-1, a+1). So for x in this interval,

        |p(x) - p(a)| = |x-a||q(x)| <= |x-a|*C     (1).

At this point it looks like delta = epsilon/C will do. But there's the
possibility that this delta > 1. That's no good, because that allows x
to wander out of (a-1, a+1), messing up (1). But if we choose delta =
min(1, epsilon/C), everything is good.


Hopefully this doesn't come off as being a bit too obtuse but
how do you get p(x) - p(a) equal to (x - a)q(x)?
I'm familiar with putting it in the following form:
|p(x) - p(a)| < epsilon and 0 < |x - a| < delta
but what polynomial does q(x) represent exactly
and how can we logically conclude that p(x) - p(a)
is equal to (x - a)q(x)?

Clearly a is a root of the polynomial p(x) - p(a), which implies (x-a)
is a factor of p(x) - p(a). Therefore p(x) - p(a) = (x-a)q(x) for some
polynomial q (of one less degree).

In your problem, p(x) = 4 + x - 3x^3 and a = 1: p(x) - p(1) = 2 + x -
3x^3, which factors into (x-1) times something. You can find the
something by polynomial division; I'm simply calling the something
q(x).
.

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