How can one calculate limits? It's important to point out that this problem is, in general, very difficult and in many cases special investigation is required for each case. However in the simple situations with which we'll be involved, some simple rules can be used. We'll summarize here the most important ones.
There is not a universal agreement on what elementary functions are. We mean here polynomials, radicals and in general powers with real exponents, trigonometric functions, logarithms, exponentials.
The main result as far these functions are concerned is that, if c is a point in the domain, we have . As we shall see this is a property of what we call continuous functions. So the problem of finding limits for these functions as x tends to a point in the domain is trivial.
Other results concerning elementary functions can be deduced by simple observation of the graphs.
Suppose we have two functions , and a real number c, that is an accumulation point for both D and E. Suppose also that (+∞ or -∞ are allowed for c, l, m). In this case it is possible to calculate the limits of the sum, the product and the quotient of the two functions, simply by summing, multiplying, dividing the two limits l and m, provided the operations between l and m are defined in the extended real line.
The only cases where the operations are not defined can be summarized as follows:
These cases require special attention. Usually they are called indeterminate forms, but we don't like this expression: there is nothing indeterminate here! The only problem is that the calculation of the limit is, in these cases, not easy. It would be better to call them difficult forms!.
Important extensions of the algebraic rules are:
, , . Observe that , as you can deduce using the graph of the function.
Sometimes functions of the form f(x)^{g(x)} are used. They are best treated, not only as far as limits are concerned, if we observe that, as a consequence of the definition of natural logarithm, we have:
.
Using this formula we can write a power were both the bases and the exponent are variables, as a power were only the exponent is variable. This proves very useful every time we want to study functions of this kind.
Observe that this formula highlights the fact that in the function f(x)^{g(x)}, the bases, f(x), must be strictly positive.
Suppose that we have three functions f, g, h, and a real number c that is an accumulation point for the domains of each function. The functions verify the following: f(x) ≤ g(x) ≤ h(x), for every x, except possibly c, in a neighbourhood of c. Suppose furthermore that . In this case we also have: .
The proof of this theorem is very easy. The following picture is self evident: if the green graph is compelled between the yellow and blue ones that pass through (0,1), it must also pass through the same point.
This theorem is known in Italy as "Teorema dei due carabinieri"
The sandwich theorem, is immediately used to prove that:
by far the most important limit in elementary analysis. You can also seee a graphical proof.
Another very important limit in elementary analysis is the following:
whose proof is omitted.