Unformatted text preview: ientl y c lose means as c lose as necessar y .
a s close
If we want 0 < C f(x)− 2 aC < ε some discrete infintesimal as small as we like,
we can choose x sufficiently close to a. For example, we may choose :
ε
0 < C x− a C < δ , w here δ = −
2
Work it out:
ε
ε
ε
0 < C x− a C < δ ⇒ 0 < C x− a C < − ⇒ a −− < x < a + −
2
2
2
ε ) + a = 2a + −
ε
f(x) = x + a = ( a + −
 2
 2
ε
ε
Now: 0 < C f(x) 2a C = C (2a + − )  2a C = − which is < ε
 2
2
ε
So when C x  a C < − we will have C f(x)  2a C < ε
2
We can say: Limit f(x) = 2a
x→a Instead of saying: Limit f(x) which is Limit (x + a)
x→a x→a we can use the infinitesimal δ x and say: Limit (x + a) = L imit ((a δ x ) + a ) = 2a x → a− δx → 0 and Limit (x+a) = L imit ((a+ δ x)+a) = 2a
x → a+ δx → 0 35 (from the left )
left
lef
(from the right)
right Let us now look at the example 9 of the previous chapter.
Example 9 :
+1 if x is rational
rational
f(x) = {
0 if x is ir r ational
irr
ir
Limit f(x) = b . Let us try to choose an ε such that C f(x) bC > ε
Suppose the x → a
for some x very close to a, that is to say for 0 < C x − a C < δ .
Recall what we said about the rationals and irrationals. Between any two rationals
there are infinitely many rationals. And between any two rationals there are also
rationals
irr
infinitely many ir r ationals .
ir
No matter how close x is to a there will always be infinitely many rational
rational
numbers and infinitely many ir r ational numbers. So we can see that as x TENDS
irr
ir
TO a, the function f(x) will keep fluctuating : f(x) will be 0 or 1.
There are two possibilities : b = 0 or b ≠ 0.
Case b = 0 : choose ε = 1/2
No matter how close x is to a , that is to say no matter how small
the δ w e choose, there will always be some r a tional n umbers
such that 0 < C x− aC < δ .
If x is rational then C f(x) bC = C 1 0C = 1 > ε = 1/2 .
So b = 0 cannot...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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