In the same way as it seems very difficult for some people to accept that 0.999... = 1, it appears that likewise some people have really hard time accepting that 00 (ie. zero to the power of zero) is indeterminate, rather than 1.
No matter how you try to twist it, the expression 00, in itself, is indeterminate in mathematics. In other words, it has no value. The expression is, in some sense, invalid. It simply has no meaningful value of any sort.
The reason why some people are so adamant that it must be 1 is that if you take smaller and smaller positive values of x, the expression xx approaches 1. In other words, the limit of xx, as x approaches 0 (from the positive side) is indeed 1.
However, limits are not the same thing as the value of the function at that point. Just because the function might approach a value as the variable approaches certain other value, doesn't necessarily mean that the function itself has that value at that point. (With many functions this is the case, but there are functions where this is not the case.)
The question of the value of 00 comes most often indeed with limits, as that's sometimes the result of certain functions. And the importance of it being indeterminate is especially large in these situations, because you can't simply assume that the answer is 1.
It can be tempting to think so, however. There is, in fact, a very large family of contiguous functions for which, taking two such functions, f(x) and g(x), which themselves approach 0 when x approaches some given value, the function f(x)g(x) does indeed approach 1 when x approaches that value.
You might try things like x2x, or sin(x)cos(x)-1, or a myriad of other combinations of functions which approach 00 when x approaches a certain value (0 in these examples), and they all indeed give 1 as result. (It can in fact be proven that this is the case with all functions that have certain smoothness characteristics.) It can thus be tempting to generalize and think that this is always the case.
However, there is a reason why 00 is considered indeterminate, not 1. There are, of course, counter-examples.
For instance, the function x1/ln(2x) approaches 00 when x approaches 0 (from the positive side), but the limit of the function is not 1, but rather e.
For another example, consider (e-x-4)x2, which approaches 00 when x approaches 0 (from either side). Its limit is 0 (rather than 1).
There are, of course, infinitely many such counter-examples, but just one is enough to demonstrate that the temptation of assuming that all such limits approach 1 will lead to incorrect results.
Yet, like with the 0.999... = 1 case, some stubborn people will try to invent new mathematics to try to make 00 always equal 1.
No matter how you try to twist it, the expression 00, in itself, is indeterminate in mathematics. In other words, it has no value. The expression is, in some sense, invalid. It simply has no meaningful value of any sort.
The reason why some people are so adamant that it must be 1 is that if you take smaller and smaller positive values of x, the expression xx approaches 1. In other words, the limit of xx, as x approaches 0 (from the positive side) is indeed 1.
However, limits are not the same thing as the value of the function at that point. Just because the function might approach a value as the variable approaches certain other value, doesn't necessarily mean that the function itself has that value at that point. (With many functions this is the case, but there are functions where this is not the case.)
The question of the value of 00 comes most often indeed with limits, as that's sometimes the result of certain functions. And the importance of it being indeterminate is especially large in these situations, because you can't simply assume that the answer is 1.
It can be tempting to think so, however. There is, in fact, a very large family of contiguous functions for which, taking two such functions, f(x) and g(x), which themselves approach 0 when x approaches some given value, the function f(x)g(x) does indeed approach 1 when x approaches that value.
You might try things like x2x, or sin(x)cos(x)-1, or a myriad of other combinations of functions which approach 00 when x approaches a certain value (0 in these examples), and they all indeed give 1 as result. (It can in fact be proven that this is the case with all functions that have certain smoothness characteristics.) It can thus be tempting to generalize and think that this is always the case.
However, there is a reason why 00 is considered indeterminate, not 1. There are, of course, counter-examples.
For instance, the function x1/ln(2x) approaches 00 when x approaches 0 (from the positive side), but the limit of the function is not 1, but rather e.
For another example, consider (e-x-4)x2, which approaches 00 when x approaches 0 (from either side). Its limit is 0 (rather than 1).
There are, of course, infinitely many such counter-examples, but just one is enough to demonstrate that the temptation of assuming that all such limits approach 1 will lead to incorrect results.
Yet, like with the 0.999... = 1 case, some stubborn people will try to invent new mathematics to try to make 00 always equal 1.
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