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

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

Pick up a high school mathematics textbook today và 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 ređiện thoại tư vấn the rule for dividing powers with like bases:

xa/xb = xa-b (x not equal to lớn 0) | (1) |

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

x0 = 1 (x not equal to lớn 0) | (4) |

Since the definition

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

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

*indeterminate*expressions in mathematics. It is not possible lớn assign a value to 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*) and *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 và that the limit can have various values of *f* & *g*. This begs the question: are these the same? Can we distinguish 00 as an indeterminate form & 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 & 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; và, if we proceed according lớn 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 lớn 1, however great or small the value of the number *a* may be, & even when a is nothing; that is to lớn say, *a*0 is equal lớn 1.

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

*az*where a is a constant & the exponent

*z*is a variable .... If

*z*= 0, then we have sầu 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.* 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 to lớn a possible problem with 00.