I am currently reading the book ''The Outer Limits of Reason'' and encountered a description about which I am very confused. I am afraid to say, this may be due to the fact that I am not a native English speaker.

on pp.61, it says:

Again there is a mathematical analogy to this. In calculus we say that the limit of 1/x as x goes to infinity is zero. That is, the larger x gets, the closer 1/x gets to zero. Since infinity is not a number, x can never get to infinity and 1/x can never get to zero. But the concept of a limit makes it meaningful. Similarly, the distance between Achilles and the Tortoise will never really be zero but the limit of the distance does get to zero. Again, we can find fault with this analogy. The concept of a mathematical limit is a type of trick. For no finite number will 1/x actually equal zero and at no time period will Achilles actually reach the Tortoise

and on pp.191, it says:

The Mandelbrot set is an easily describable set of complex numbers. Start with a complex number c, square it, and add c to the result. This gives you another complex number. Square this number and add c to it again. Iteratively continue this procedure over and over again—that is, take the complex number z, calculate z2 + c, and repeat.5 Either of two things can happen to the numbers in this iteration:

• They can get bigger and bigger until they go off to infinity.

• The complex numbers can remain small.

I think I know that when we say "as x goes to infinity" it means that "as x is moving towards infinity", right? But I wanted to know is "goes off to infinity" means the same thing or means that "it reaches to infinity"? or it means something different?

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    $\begingroup$ consider $\lim\limits_{x\to\color{blue}\infty}\dfrac1x=0$ vs. $\lim\limits_{x\to0}\dfrac 1x=\color{blue}\infty$ $\endgroup$ – J. W. Tanner Mar 18 at 13:41
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    $\begingroup$ They mean the same thing. $\endgroup$ – NickD Mar 18 at 13:41

In standard calculus, nothing ever reaches infinity. However, they can grow without bound. There are many phrases we use to describe this phenomenon, because it's a very common thing and our books would get very monotonous if we used the same phrase every single time.

Yes, they all mean the same.

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