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Why does one have to prove that the limit below exists, in order to prove the prime number theorem? $$ \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} $$ Doesn't the fact that the Chebyshev function is monotonic imply that it does exist? If it doesn't then how does one prove that the limit exists? How would I even try to figure out what the limit superior and inferior are?

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    Explain your notation. What is $\psi (x)$ here? Is it Chebyshev function?2012-11-13
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    ${e^x}$ is monotone but ${\lim _{x \to \infty }}{e^x}/x = \infty$.2012-11-13
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    Chebyshev function is not continuous, so lim sup and lim inf are not defined here.2012-11-13
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    @glebovg: they may be $+\infty$ or $-\infty$, but $\limsup$ and $\liminf$ always exist.2012-11-13

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