Suppose f and g are two functions, and we are interested in what happens to $f(x)$ and $g(x)$ as x grows. We say that " f is asymptotic to g ", or $f\sim~g,$ if their ratio tends to 1. For instance, $x^2+5x-\log~x\sim~x^2,$ and (more interestingly and much less obviously) $\pi(x)\sim\text{Li}(x)$ where $\pi$ is the prime counting function and Li is the logarithmic integral.

This last is the Prime Number Theorem.

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