Unformatted text preview: be the Limit f(x) .
x→a Case b ≠ 0 : choose ε = C 1  b C
choose
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  bC > ε = C 1  b C .
2
So b ≠ 0 cannot be the Limit f(x) .
→
x a Since a can be any point on the real number line, Limit f(x) does not exist anywhere.
x→a
36 10. ALUE
V AL UE v er sus LIMIT So far when we took Limits of functions as x → a or δ x → 0 we ASSUMED
that these Limits exist. But this is not so simple.
1. Does the Limit exist ?
2. Is the Limit from the right = Limit from the left ?
i.e. is L imit f(x) = L imit f(x)
x→a x>a
right
3. Is the x → a+ x<a
left Value f(x) = L imit
x→a
x=a f(x) In finding the Limit w e first simplify and then put x = a or x = 0 or whatever
and evaluate. We can simplify because:
δ x only TENDS TO 0, δx ≠ 0 ,
x only TENDS TO a, x ≠ a ,
x only TENDS TO 0, x ≠ 0 .
Whereas in finding the Value w e must directly put x = a or x = 0 or whatever
and evaluate. 37 Let us compare Value f(x) and L imit f(x)
x=a We have the following possibilities: x→a f(x) = x (i) Value and Limit both exist and are equal :
(ii) Value and Limit exist but are not equal.
Let f(x) = { L imit x → a 2 for x = a
1 for x ≠ a imit
f(x) = 1 = L→ + f(x) , Value f(x) = 2
x a x=a Try drawing the graph of this function.
(iii) Value exists but Limit does not.
Let f(x) = 1  x for x ≤ 1 { 3  x for 1 < x L imit f(x) = 0 ≠ L imit f (x) = 2 , Value f(x) = 0
x → 1+
x=1 x → 1 Try drawing the graph of this function.
(iv) Value does not exist but Limit does.
Let f(x) = x2  a2 , L imit
x  a
x→a f(x) = 2a, Value
x=a f(x) = ? L
Let f(x) = sin x , x imit f(x) = 1, Value f(x) = ?
→0
x
x=0
(v) Value and Limit do not exist.
1
Let f(x) = x , L imit f (x) = ? , Value
x→0
x=0
38 f(x) = ? 1 1. CONTINUITY of a Function Our common perception of continuous is so taken for granted that we se...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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