# How to determine how fast something is going towards infinity?

Just for a little bit of information I'm more of a programmer and less a mathematician so if some of my terms seem out of place it is due to a lack for formal training in Math.

While working on my hobby which I posted a question about it here. Information on this fractal?

I was trying to figure out how fast a recursive function with specific inputs would be reaching infinity compared to the rest of set.

Right now my method is very odd. I count the number of iterations I can do before my program fails and throws an error. I catch this error and I use the iteration count as my metric. The lower the iteration count the faster the value tends towards infinity.

I was hoping for a more mathematical way to tackle this rather than just letting my program run till it fails (note if it reaches a max iterations value I just label it as not tending towards infinity).

The recursive set of equations I'm using are the same as the link but I'll just copy and paste them bellow for ease.

$$a_{i+1} = (a_i - b_i) / c_i$$ $$b_{i+1} = (b_i - c_i) / a_i$$ $$c_{i+1} = (c_i - a_i) / b_i$$

a,b and c are all real numbers.

For answers assume my knowledge of math is that of a senor in high school or first year university. It's been well over a decade that I've taken any university math classes so I have to google most words/terms/symbols used in answers here.

Note: Helping with correctly tagging this questions would also be appreciated.

• Your approach is not so weird at all. For the classic Mandelbrot set you can show that once the magnitude of the point exceeds $2$ it will go off to infinity. I believe many of the programs stop when the magnitude exceeds $2$ and plot the number of iterations as the dwell time. They also have a maximum iteration count where it it has not exceeded a magnitude of $2$ by that count they tag it as never going to infinity. – Ross Millikan Feb 14 at 22:15