In $7^{th}$ grade, in order to learn divisibility, memory, and focus, my math teacher had my pre-algebra class play a game called Frazzle. To play the game Frazzle, each person went around the room and said the next number (someone would say 1, then 2, etc), but if the number had a 7 in it (such as 721) or was divisible by 7 (such as 21), you yelled frazzle. Admittedly the game was a blast, but I've always had this question in my mind. In the game, I found that as we continued into 2 or 3 digit numbers, the word frazzle came up much more. For example in 1-->10, 1 number frazzle. 1-->20, 3 numbers would frazzle. I conjecture that: $$\lim_{n\to \infty}f/n=1$$ In other words that 100% of the numbers would frazzle, with n standing for the number that you are on and f being the number of frazzled numbers. But that creates a conundrum 1 is not divisible by 7. or 2.
-Please either provide a proof or disproof of what I am saying or disprove it. I find myself extremely confused at this odd paradox. Thanks loads.
Let’s look at the non-negative integers having base ten representations requiring at most $d$ digits. (Including $0$ makes no different in the limit and makes the calculation simpler.) There are $10^d$ of them. Of those, $9^d$ have representations without a $7$. Thus, the fraction of these numbers with a $7$ in their representations is
$$\frac{10^d-9^d}{10^d}=1-\left(\frac9{10}\right)^d\;,$$
which approaches $1$ as $d$ increases without bound. Thus, your conjecture is correct, even without including the multiples of $7$.