Unformatted text preview: at x TENDS TO 0.5, we write this as: x → a .
Will x = a ? No. We will always have 0 < C x −a C
L I M I T:
If we let b = 0.51 we are correct in saying: x TENDS TO 0.51, denoted by x → b.
Will x = b ? No. 0 < Cx−bC
What is the difference between x → a and x → b ?
In x → b : we cannot choose x as close as we like to b . The difference
as close we like to
we like
as
C x− bC c annot be made as small as w e lik e . We cannot have C x −bC < δ
for any small quantity δ as small as we like . For example:
as
if we choose δ = 0 .001 we will have 0 < δ < 0 .01 < C x − bC . 14 In x → a : we can choose x as close as we like to a. The difference C x− aC c an
as
to
be made as small as we like . If we choose δ > 0 , any small positive
as
quantity as small as we like , we can choose x such that C x − aC < δ .
as
Since we can choose x as close as we like to a ,
as
to
and only to a but not to any other point , we say:
p oint
LIMIT as x → a i s a , written as L imit x = a
x→a
In x = 0.4, 0.49, 0.499, 0.4999, ... progressively with a = 0.5, since x < a, we
can be more precise and say x → a from the left. We write this as x → a.
We could have let x = 0.51, 0.501, 0.5001, 0.50001, ... progressively with
a = 0.5 . Here too we have x → a but from the right We write this as x → a + .
right.
Sometimes we will be very specific and say:
x TENDS TO a from the left written as : x → a left,
x TENDS TO a from the right written as : x → a+
right,
Which ever be the case, whether x → a+ ( from the right or x → a  (from
right)
the left we insist that x ≠ a, i.e. 0 < C x−aC . The case when x = a we will
left)
speak of separately. L
Since we know that LIMIT (x → a − ) is a , we write : x imita { x } = a .
→
Since we know that LIMIT (x → a + ) is a , we write : L imita+ { x } = a .
x→
x = 0.4, 0.49, 0.499, ... ...,0.5001, 0.501, 0.51= x
a +→ x
x → a a=0.5
−
3 −
2 −
1 0 15 1 2 3 5 . INFINITESIMALS
Let us now focus on the dif...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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