In the last post, I said that I would do a post on what zero divided by zero was. Today, we’re covering that!
Credit to Eddie Woo, the person who (through YouTube) provided the information. The video is linked here.
In maths, a limit is the value that infinitely approaches the x value but never reaches it. For example, the limit of x/7 as x reaches 7 is 1. Sometimes, the value will exist, so the limit could be the value. Limits are a fundamental part of calculus, as calculus is about infinitesimal changes and such.
In Eddie Woo’s video, he drew a table on the board, descending from 1 all the way to 0.1. He assigned some of his students the numbers, and told them to evaluate x^x, where x was their assigned numbers. The numbers were (rounded to three decimal points):
- 1^1 = 1.000
- 0.9^0.9 = 0.910
- 0.8^0.8 = 0.837
- 0.7^0.7 = 0.779
- 0.6^0.6 = 0.736
It is evident that as the value of x decreases, so does x^x. Let’s keep going!
- 0.5^0.5 = 0.707
- 0.4^0.4 = 0.693
- 0.3^0.3 = 0.696
And… what? How was it that for 0.4, x^x was 0.693, but for 0.3, x^x was 0.696? It increased by 0.003!
I guess… let’s keep going.
- 0.2^0.2 = 0.724
- 0.1^0.1 = 0.794
We’ve reached the end, but let’s go further.
- 0.01^0.01 = 0.955
Already, we’ve reached 0.9, but let’s go further. 0.001^0.001 is 0.993, 0.0001^0.0001 is 0.999. That’s almost 1! If we keep going, we’ll get closer and closer to 1, until about sixteen digits, where the calculator freaks out and says 1.
Putting the function y = x^x into a graphing calculator, it actually shows that 0^0 is about 1.
Now, why is that?
You see, for roots of a decimal lower than 1 but higher than 0, it has to actually be higher than said decimal. For example, the square root of 0.5 is about 0.707, as 0.7*0.7 is 0.49. And as the root gets higher, so does the value. For example, the square root of 0.1 is only 0.316, while its tenth root is 0.794.
In fact, the turning point is about x = 0.368, thanks to good old Desmos.
This was a great post to write, and maybe I’ll do an extension post to this in the future.
Until then, peace.