1 Introduction
Let’s start working mathematically on limits of functions.
2 Basic Properties
A nice property of limit is that it is linear:
In fact, one important spirit of calculus (differentiation + integration) is to turn nonlinear things into linear ones. Even better, limit commute with all elementary algebraic operations:
We take for granted, the nice things, that for all the elementary functions in- troduced in Note 1,
x→climf (x) = f (c)
if c is in the domain (i.e. f is defined at c). This can be a problem if we working on function of the formp(x)q(x)because the q(c) can be 0. Here are the possibilities:
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So the only interesting limits for p(x)q(x) at c are when both the top and bottom approach 0 as x → c. We compute them case-by-case using elementary algebras:
For the case c = ∞, the limit only depends on the terms that dominate:
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3 The Squeeze Theorem
A very important theorem on limit is the squeeze theorem. This is quite intu- itive: if the big function and small function both approach the same limit the function in between must approach that same limit:
Here are some simple consequences:
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A more important application is the following limit identity, which is used to compute the limits involving trigonometric functions:
Theorem 3.1.
lim
x→0
sin x x = 1.
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Let’s discuss some examples following this formula:
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4 Continuity
We conclude this chapter by the concept of continuity. In plain words, it says that functions can not jump. In another words, we can always keep f (x) as close as we want, as long as we keep x close:
Let’s re-visit the examples we have discussed at the beginning:
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All elementary functions mentioned in note 1 are continuous as long as we do not have 0 denominator. The coming chapter, the differentiation, will give us ways to distinguish these continuous functions:
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