| Fig. 10. |
38. An important application of the theorem of intermediate value and its generalization can be made to the problem of evaluating certain limits. If two functions φ(x) and ψ(x) both vanish at x = a, the fraction φ(x)/ψ(x) may have a finite Indeterminate forms. limit at a. This limit is described as the limit of an “indeterminate form.” Such indeterminate forms were considered first by de l’Hospital (1696) to whom the problem of evaluating the limit presented itself in the form of tracing the curve y = φ(x)/ψ(x) near the ordinate x = a, when the curves y = φ(x) and y = ψ(x) both cross the axis of x at the same point as this ordinate. In fig. 10 PA and QA represent short arcs of the curves φ, ψ, chosen so that P and Q have the same abscissa. The value of the ordinate of the corresponding point R of the compound curve is given by the ratio of the ordinates PM, QM. De l’Hospital treated PM and QM as “infinitesimal,” so that the equations PM : AM =φ’(a) and QM : AM = ψ′(a) could be assumed to hold, and he arrived at the result that the “true value” of φ(a)/ψ(a) is φ′(a)/ψ′(a). It can be proved rigorously that, if ψ′(x) does not vanish at x = a, while φ(a) = 0 and ψ(a) = 0, then
| lim.x=a | φ(x) | = | φ′(a) | . |
| ψ(x) | ψ′(a) |
It can be proved further if that φm(x) and ψn(x) are the differential coefficients of lowest order of φ(x) and ψ(x) which do not vanish at x = a, and if m = n, then
| lim.x=a | φ(x) | = | φn(a) | . |
| ψ(x) | ψn(a) |
If m > n the limit is zero; but if m < n the function represented by the quotient φ(x)/ψ(x) “becomes infinite” at x = a. If the value of the function at x = a is not assigned by the definition of the function, the function does not exist at x = a, and the meaning of the statement that it “becomes infinite” is that it has no finite limit. The statement does not mean that the function has a value which we call infinity. There is no such value (see [Function]).
Such indeterminate forms as that described above are said to be of the form 0/0. Other indeterminate forms are presented in the form 0 × ∞, or 1∞, or ∞/∞, or ∞ − ∞. The most notable of the forms 1∞ is lim.x=0(1 + x)1/x, which is e. The case in which φ(x) and ψ(x) both tend to become infinite at x = a is reducible to the case in which both the functions tend to become infinite when x is increased indefinitely. If φ′(x) and ψ′(x) have determinate finite limits when x is increased indefinitely, while φ(x) and ψ(x) are determinately (positively or negatively) infinite, we have the result expressed by the equation
| lim.x=∞ | φ(x) | = | lim.x=∞ φ′(x) | . |
| ψ(x) | lim.x=∞ ψ′(x) |
For the meaning of the statement that φ(x) and ψ(x) are determinately infinite reference may be made to the article [Function]. The evaluation of forms of the type ∞/∞ leads to a scale of increasing “infinities,” each being infinite in comparison with the preceding. Such a scale is
log x, ... x, x2, ... xn, ... ex, ... xx;