I was playing around with my calculator before finding out it has a graph feature, I inputted X^∞ and it came up with a straight line from -1 to 1, Im curious as to why it does this.
In: 2
Take a few scenarios. Starting at x^1 and then going to x^2 then x^10 and so on.
You can see that anything above 1 or below -1 quickly grows in magnitude off towards an infinity. Anything between -1 and 1 approaches zero. Taken to the extreme, we’d expect y=0 when -1<x<1. The result is a line where y=0 from x=-1 to 0.
When you are multiplying numbers below 1 and above -1, the result gets closer to 0. So when you do this an infinite number of times, the result will be 0.
Thus the graph should go from x= -1 to 1 with 0 on the y axis.
any number between -1 and 1 gets smaller when multiplied by itself. if you were to multiply any of these numbers by themselves infinitely many times, you would get zero. so the result is a flat line with height at zero
What does 2^∞ means? It’s the answer to the question “If a population doubles every day, then how many peoples will there be at the end of time?” which is an infinite amount of peoples, so not something you can put on a drawing
What does (1/2)^∞ means? It’s the answer to the question “If a population halves every day, then how many peoples will there be at the end of time?” which is 0.
More generally:
*For x>1 then x^∞ =+∞, so not drawn.
*For x=1 then x^∞ =1, but your calculator might not be displaying it correctly.
*For -1<x<1 then x^∞ =0, which is the straight line you see.
*For x=-1 things are weird since x^∞ is equal to both -1 and +1 at the same times. Your calculator is probably considering that this is “undefined”, so not drawn.
*For x<-1 things are still weird and x^∞ is equal to both -∞ and +∞ at the same time. But in any case it’s not drawn.
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