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Unformatted text preview: rst reading it is not necessary to study the Analysis of Limits. The reader may
skip the next chapter without loss of continuity.
9. Analysis of Limits
In general, we can talk about the LIMIT of a function f(x) as x → a .
Does f(x) TEND TO any par ticular value b as x TENDS TO a ?
Can we choose f(x) as close as we like to b b y choosing x sufficiently
close t o a ?
Does L imit − f(x) = b = L imit + f(x) ?
x→a x→a Does Limit f(a-- δ x) = b = Limit
δx → 0 δx → 0 f(a+ δ x) ? If it does, then we call this b the LIMIT of f(x) as x TENDS TO a and we write
L imit f(x) = b or L imit f(a+ δ x) = b
δx → 0
We are not concerned with if 0 < C f(x)-- bC o r 0 = C f(x) -- b C
What we are concerned with is when x → a , for any discrete infinitesimal
ε a s small as we like :
can we have C f(x) -- b C < ε o r equivalently C f(a + δ x) -- b C < ε ?
Again, we stress that x = a. f(x) at the instant x = a as we said earlier, is
Value f(x) = f(a), if it exists.
x=a We may combine x → a− and x → a+ into one expression : C x -- aC < δ .
Formally, we may define the LIMIT of a function f(x) as x TENDS TO a : Limit f(x) = b if for any discrete infinitesimal ε
as small as we like, we can find δ such that :
C f(x) -- b C < ε when C x -- aC < δ .
*Note : A more detailed and rigorous presentation is in A LITTLE MORE CALCULUS by the author. 34 Based on the formal definition of the “limit of a function” , we may analyse the
behaviour of a function near some instant a, and see if it has a Limit or not. Let us
look at the example 2 of the previous chapter.
Exampl e 2: Consider the function f(x) = x + a
What is the L imit o f f(x) as x TENDS TO a ? As x → a w e have f(x) → 2 a
Since 0 < C x− a C w e have 0 < C f(x)− 2a C , i.e. f(x) = 2 a.
We can choose f(x) as close as we like to 2a by choosing x suf f iciently
as close we like to
uff iciently close
c lose t o a. S uf f ic...
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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