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I have seen how to solve some limits, including how to solve analytically for the limiting behavior of a series, however, I have encountered some limits that I have not been able to solve like the definition of $e$ or the Riemann for the area under $\sqrt{1-x^2}$, am I missing the techniques required to solve them analytically or is there something about transcendental numbers that does not allow these limits to be solved?

GuPe
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First, we should define what we mean my being "analytic". If it is used to mean "may be expressed with elementary functions finitely", then such solutions are possible only in some simple cases. In general, approximate, numerical methods need to be used. If by "analytic" we mean "may be expressed as a power series", then the class of such solutions is broader but still not includes all the possible solutions of limits, series, definite integrals etc.

Generally speaking, all those problems are formulated using functions, $\mathbb{N}\to\mathbb{R}$ or $\mathbb{R}\to\mathbb{R}$, and operations with functions, without knowing their exact definitions, are very restricted: there are many logical obstacles. See "Constructive Mathematics", by the way.

So, only quite few (but important) special cases may be solved analytically (whatever it means), all the rest need a "brute force" numerical approach. But, before the computations may be performed, the existence of the solution(s) must be established.

P.S. There is no algorithm to even know if a two given expressions represent the same function. That's the reason why there is a problem with real numbers (they are functions, thus, cannot be compared in general).

The ability to express a result finitely depends on the set of functions allowed, whose properties are established. And there is no finite set of such functions to express say any series finitely, as just a composition. There are however some proofs that some particular values may not be finitely expressed through a particular set of "elementary" functions.

Shortly, no, it's all knowledge + creativity. See also "Special Functions".

Andrew
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  • Is there a way of knowing which limits must be solved numerically? – GuPe Feb 26 '16 at 15:47
  • Shortly: no. Every limit must be solved numerically unless someone discovered (proved) a nifty expression for it. The general "heuristic" algorithm is to try to identify if a given limit (problem) falls into a known class (known to you), if not - then go numeric. – Andrew Feb 27 '16 at 05:21