Unformatted text preview: ference ∆ = C a−x C as x → a = 0.5, be it from
the left or from the right.
left
right
∆ = C a−x C = 0.01, 0.001, 0.0001, 0.00001, . . .
After a while it becomes infinitely small and we denote it by δ. We may say :
infinitel
infinitely
1
δ = /10n as n → ∞
1
We can see that LIMIT { /10n } = 0 .
n→∞
Let us look at another example. Imagine A and B located one meter apart on the
xaxis . At each second let B jump 1/2 distance towards A. We may say that B → A.
/
/
A ← 18 14
a 1
/2 B xaxis a + 1 meter The difference ∆ = C A − BC = 1 2 , 1 4 , 1 8 , . . .
///
After a while it becomes infinitely small and we denote it by δ. We may say :
infinitel
infinitely
1
δ = /2n as n → ∞
1
We can see that LIMIT { /2n } = 0 .
n→∞
By definition:
An infinitely small q uantity whose LIMIT is zero is
infinitely
called an INFINITESIMAL.
Note that the LIMIT is zero, i.e. the quantity δ TENDS TO zero but
T ENDS
does not become z er o. δ does not v anish.
zer
ero
vanish.
d oes
16 We may now use the infinitesimal δ and say:
x → a+ ≡ x = a + δ
x → a ≡ x = a − δ
B→A ≡ B=a+δ
To express the concept x coincides with a, we may say :
from the right : x = Limit { a + δ }
right
δ→0
from the left : x = Limit { a − δ }
l eft
δ→0
Likewise, to express B coincides with A we may say : B = Limit
{a+δ}
δ→0 As B jumps from 1 to 1 2 to 1 4 to 1 8 . . . to A it skips a lot points in between.
/
/
/
points
Likewise, as x → a it skips a lot of points in between. This is because δ is a
points
discrete infinitesimal .
We can think of the number line or xaxis as smooth and CONTINUOUS.There
are no gaps or breaks. In Calculus we want to study the behavior of a function
at each and every point on the xaxis or instant on the TIME AXIS.
So we want x to TEND TO a point in a smooth and CONTINUOUS manner.
For the infinitesimal that TENDS TO zero in a smooth and CONTINUOUS
manner we have the special notation δ x.
Geometrically, we can think of δ x as a small CONTINUOUS line segment
representing an infinitely small CHANGE in x. The length of this small line
segment δ x is always > 0 no matter where we are on the xaxis. δ x gets
smaller and smaller. The length of δ x TENDS TO zero but does not become
zero. δ x does not vanish. However, the LIMIT of δx is zero. 17 We may let : 1
δx = /10x as x → ∞
1
or δx = /2x as x → ∞
Now we may express x → a in a continuous manner by :
continuous
x → a+ ≡ x = a + δx
x → a...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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