Where's the line between persistent and desperate?

I've seen a lot of answers from girls stating that they want a guy to be persistent and chase them, yet won't chase guys for fear of looking 'desperate' and 'clingy'. However, guys, for the most part, have this fear that chasing girls will be construed as desperate and clingy as well. Where is that fine line between being persistent and overly-persistent or desperate? What would be some behavior s characteristic of an 'attractive' level of persistence versus an 'unattractive' level?
Where's the line between persistent and desperate?
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