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.
33 Limits*
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
as
s ufficiently
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
this as:
L imit f(x) = b or L imit f(a+ δ x) = b
x→a
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
discrete infinitesimal
discr
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
/
i nstant
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
discrete
x→a
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
“limit
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.
Example 2:
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
sufficiently
as close we like to
s uf
uff iciently close
necessary
c lose t o a. S uf f ic...
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 Fall '09
 TAMERDOğAN

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