When calculus books state that 00 is an indeterminate size, they mean that there are functions f(x) và g(x) such that f(x) approaches 0 & g(x) approaches 0 as x approaches 0, & that one must evaluate the limit of

When calculus books state that 00 is an indeterminate size, they mean that there are functions f(x) & g(x) such that f(x) approaches 0 and g(x) approaches 0 as x approaches 0, and that one must evaluate the limit of

Piông xã up a high school mathematics textbook today & you will see that 00 is treated as an *indeterminate form*. For example, the following is taken from a current New York Regents text <6>:

We regọi the rule for dividing powers with lượt thích bases:

xa/xb = xa-b (x not equal khổng lồ 0) | (1) |

Therefore, in order for *x*0 khổng lồ be meaningful, we must make the following definition:

x0 = 1 (x not equal to 0) | (4) |

Since the definition

*x*0 = 1 is based upon division, and division by 0 is not possible, we have stated that

*x*is not equal lớn 0. Actually, the expression 00 (0 to lớn the zero power) is one of several

*indeterminate*expressions in mathematics. It is not possible khổng lồ assign a value lớn an indeterminate expression.

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Calculus textbooks also discuss the problem, usually in a section dealing with L"Hospital"s Rule. Suppose we are given two functions, *f*(*x*) & *g*(*x*), with the properties that (lim_x
ightarrow a f(x)=0) & (lim_x
ightarrow a g(x)=0.) When attempting khổng lồ evaluate <*f*(*x*)>*g*(*x*) in the limit as *x* approaches *a*, we are told rightly that this is an *indeterminate form* of type 00 & that the limit can have various values of *f* và *g*. This begs the question: are these the same? Can we distinguish 00 as an indeterminate khung & 00 as a number? The treatment of 00 has been discussed for several hundred years. Donald Knuth <7> points out that an Italian count by the name of Guglielmo Libri published several papers in the 1830s on the subject of 00 and its properties. However, in his *Elements of Algebra*, (1770) <4>, which was published years before Libri, Euler wrote,

As in this series of powers each term is found by multiplying the preceding term by *a,* which increases the exponent by 1; so when any term is given, we may also find the preceding term, if we divide by *a,* because this diminishes the exponent by 1. This shews that the term which precedes the first term *a*1 must necessarily be *a*/*a* or 1; and, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must be *a*0; và hence we deduce this remarkable property, that *a*0 is always equal to 1, however great or small the value of the number *a* may be, và even when a is nothing; that is lớn say, *a*0 is equal to lớn 1.

More from Euler: In his *Introduction to Analysis of the Infinite* (1748) <5>, he writes :

*az*where a is a constant và the exponent

*z*is a variable .... If

*z*= 0, then we have a0 = 1. If

*a*= 0, we take a huge jump in the values of

*az*. As long as the value of

*z*remains positive sầu, or greater than zero, then we always have

*az*= 0. If

*z*= 0, then

*a*0 = 1.

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Euler defines the logarithm of *y* as the value of the function *z,* such that *az* = *y.Xem thêm: Cơ Sở Dữ Liệu Phân Tán Là Gì, Bài 12: Các Loại Kiến Trúc Của Hệ Cơ Sở Dữ Liệu* He writes that it is understood that the base

*a*of the logarithm should be a number greater than 1, thus avoiding his earlier reference lớn a possible problem with 00.