What would 0 0 with proof

Zero to zero

In the definition of the power a0 = 1 was set for all a, so is in particular
Since 0x has the value 0 for all positive x, the value 0 would also be conceivable. Like the determination that 1 is not a prime number, the determination of the value of 00 is also not a question of true or false, but of useful or inexpedient.

Historical remarks

Until the beginning of the 19th century, mathematicians apparently set 00 = 1 without questioning this definition in more detail. Cauchy[1] however, listed 00 along with other expressions such as 0/0 in a table of indefinite expressions. Apparently he wanted to point out that for every real number w≥0functions f, g can be specified in such a way that f (a) = g (a) = 0 and
limx → a f (x) g (x) = w
Limit arguments are therefore unsuitable for specifying 00.
Libri published in 1833[2] a work in which he presented unconvincing arguments for 00 = 1, which were subsequently controversially discussed. In defense of Libri, Möbius published[3] a proof of his teacher Johann Pfaff[4], which essentially showed that limx → 0 + xx = 1, and an alleged proof for limx → 0 + f (x) g (x) = 1 if limx → 0 + f (x) = limx → 0 + G (x) = 0. This proof was quickly refuted by the counterexample f (x) = e − 1 / x and g (x) = x. As a result, the controversy fell silent and the convention of leaving 00 undefined spread more and more in analysis textbooks.
Donald Ervin Knuth mentioned the history of the controversy in the American Mathematical Monthly in 1992 and firmly opposed the conclusion that 00 is left undefined. If 00 = 1 cannot be assumed, many mathematical theorems such as the binomial theorem require
(x + y) n = k = 0∑n (kn) xkyn − k
a special treatment for the cases x = 0 or y = 0 or at the same time n = 0 and x + y = 0.
ex = n = 0∑∞ n! xn
at the point x = 0 or in the sum formula for the geometric series
k = 0∑n qk = 1 − q1 − qn + 1
for q = 0. The convention 00 = 1 is also useful here.
The convention 00 = 1 is therefore useful for practical reasons, because it simplifies the formulation of many mathematical expressions. However, since this convention is not generally accepted, it is advisable to refer explicitly to the definition 00 = 1 used. 00 = 1 by definition does not mean that the function xy would be continuous at the point x = y = 0.
1 Augustin-Louis Cauchy, 1789-1857, a French mathematician.
2 Guglielmo Libri, 1803-1869, Italian-French mathematician and bibliophile.
3 August Ferdinand Möbius, 1790-1868, a German mathematician and astronomer
4 Johann Friedrich Pfaff, 1765-1825, a German mathematician

One must study mathematics because it organizes thoughts.

M. W. Lomonosov

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