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I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest root)?

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  • $\begingroup$ There are lots of known bounds on the roots of a given polynomial. See the Wikipedia article on "Properties of polynomial roots, section on "Bounds on (complex) polynomial roots". $\endgroup$ – Arturo Magidin Jan 25 '16 at 17:38
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    $\begingroup$ Is your goal numerical computation? Or do you just want to express the largest root of a polynomial as a limit for theoretical purposes (for instance, you are looking for a proof strategy)? $\endgroup$ – Federico Poloni Jan 25 '16 at 17:48
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For computing the largest root, see the Graeffe Method.

For bounding the largest root, there are many bounds, see the Wikipedia article (which is incomplete, but since you don't tell us what you want, maybe that's good enough).

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