Prove that lim x->3 x^2 = 9
when
The book factors the expression
Then it says that if we can find a positive constant C, then we can write
It then goes on to say we can make
by taking
^ What does it mean 'make
' by moving the constant over to the other side? If they wrote it like that, then doesn't it already mean that they've chosen some C such that the expression on the LHS is less than epsilon?
Also, I don't understand the motivation for finding the constant C and doing the above. Is it just to try and get the expression into the form
?
Then after that, it says we can find a number C if we restrict x to lie in some interval centered at 3. Since we are interested only in values that are close to 3, it is reasonable to assume that x is within a distance 1 from 3' - is that just a random small number they chose? is 1 the standard to be chosen in similar limit problems?
So that leads to
Then that leads to
, thus
. Does that expression mean that we only care that
is less than
, not what the valid values of x are?
So then they choose
remembering that
but there are two restrictions on
and
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So
….they show that the
works and they choose
and get
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so for the limit to work, you need to use 2 different deltas simulatenously or something? this is rather confusing as the steps seem a bit random to me
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also, when they prove
, and they use
, they did
^ is
simply because
and the inequality's just been multiplied by 4?
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