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Suppose that $E_t$ represents the highest Erdos number by year $t$. Find the following limit.

$\lim_{t \to \infty} E_t$

What do you hope this limit is? What is it really?

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    What is Erdos number ? – 2017-02-17
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    I think if you don't publish, you have an undefined Erdos number. – 2017-02-17

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I don't know how to answer your first question, but I believe the limit will approach $\infty$.

The Erdős number measures how close an author has been to publishing a paper with Paul Erdős. For example, if I publish a paper directly with Erdős, I have an Erdős number of $1$. If I publish a paper with someone who published a paper with Erdős directly, I have an Erdos number of $2$.

Since Erdős died in 1996, no one since then can directly publish with Erdős, so an Erdős number of $1$ has been impossible since 1996 (except for papers published after his death).

Over time, all authors with Erdős number of 1 will die, so no one after that last death could achieve an Erdős number of 2.

Once the last person with Erdős number $n$ dies, no one could achieve an Erdős number of $n+1$.

Therefore, $$\lim_{t\to\infty} E_t=\infty\ .$$

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    This answer cheerfully assumes the continued existence of the human race forever. (Perhaps this is what OP meant about hope) – 2017-02-17
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    On the other hand it also assumes there will never be immortality, so maybe it's a wash. – 2017-02-17