Real Analysis – Limits and Continuity III
The first thing I want to say is that the concept of limit is a local concept.
In mathematical lingo what this means is that for a function to have a limit in a given point, , it doesn’t matter how the function behaves when we are far away from the point in question, what matters is just how the function behaves in the vicinity of the point.
This all very good for common day to day knowledge but it is not good enough for Mathematics.
So, with the concept of limit what we are doing is formalizing what we mean with the expressions far away and vicinity.
For an example let me introduce the function
This function’s not a very sophisticated but it’s good enough for what I’m trying to convey.
First of all let us plot this function to see what it looks like.
Where we have drawn the case in blue and the case in red.
It is easy to see that for all different than the function has no limit.
For and . Thus , and we can conclude that this limit doesn’t exist.
For it is possible to prove (I’ll do that when the concept of limit is formalized using an condition) that .
They don’t make concepts more local than this! This function only has a limit at point .
In an intuitive way we can understand this result like this: we can think that the concept of limit is a measure of how good behaved a function is.
Since this function is always jumping from point to point as we move from rational numbers to irrational numbers we can say that it isn’t a well behaved one.
The former statement is true almost everywhere in the domain of the function. The only point where it breaks down is at point .
This is so because even though the function is a badly behaved one it misbehaves less and less while .
Theorem 32 Let , , ; and let us suppose that there exists such as . Then, if it also is . And if it also is
Proof: Omitted. |
The previous theorem states a very straightforward fact, but, as always, what matters is that this result can be proven. In more prosaic terms this theorem expresses the conditions that need to fulfilled for us to know the limits of some functions just by knowing the limit of another one.
It may well be the case that one limit may be very easy to calculate while the other is not.
If we can establish an order relationship and calculate one of the limits it is possible for us to conclude something about the limit of the other function. In Theorem 32 we where particularly interested in the cases when the limit is but we already seen in Theorem 30 that limit weakens order relationships.
In this case if it is , for some neighbourhood around a point , then we know that also. Now, if has no choice but to go to positive infinity as we move closer to since it has to be larger than .
In the case a similar reasoning applies.
is smaller than and if gets to smaller and smaller values as we approach than also has to get smaller and smaller values.
Theorem 33 (Squeezed function theorem) Let , , ; and let us suppose that there exists such as . If it also is . Proof: Omitted. |
This theorem continues the trend of computing limits of functions without computing them!
In here if we can box the function in a neighbourhood of a point by two functions, and if we compute the limits of the boxing functions and come to the conclusion that they are equal we are able to know that the boxed function has the same limit.
As an example let us see the limit:
It is . Thus .
Since it also is .
As a second example let us now look into:
Since
It is and . Thus it also is
— 6.5. Algebraic Properties of the Limit of Functions —
Just like we did for sequences we’ll derive the algebraic rules that allows to manipulate the limits of some more complex expressions.
April 23, 2011 at 3:51 pm
[…] As a final example let us look at the modified Dirichlet function that was introduced at this post. […]
March 19, 2014 at 4:41 am
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