A colleague has been frustrated by the apparent limitations to Maple's abilities to "solve" inequalities. This does appear to be something that should - and could - be improved with a little effort.
The typical problem under consideration is the epsilon-delta definition of limit. Ideally, it would be nice to execute a command such as
> solve( abs( f-L ) < epsilon, x );
and receive an answer in terms of intervals.
If epsilon is a name (not assigned a numerical value) this might be restricted to a relatively small class of functions. (At present it does not work even for a linear function. It's a little better without the abs but not much.)
If epsilon is numeric (and if f and L do not involve unspecified parameters), a numerical approximation should be possible. Even more should be possible if the user provides an interval on which to look.
As I discussed this with my colleague, it dawned on me that much of what I am requesting here mirrors the development of the fsolve command. Recent releases of Maple have seen dramatic improvements to fsolve, in particular the interval, starting_values, avoid, and maxsols options.
A typical problem that one would like to be able to solve numerically would be
> fsolve( abs( sin(x)/x-1 ) < 0.01, x=-1..1 );
I am well aware of ways to obtain this information using plot and fsolve on restricted intervals. What I believe should be possible - and would like to see in the next release of Maple - is the natural support for problems of this type (for a reasonable class of functions).
If I have overlooked something in Maple 10 that provides support for this type of problem please let me know.
As a side note, it would be nice to have Maple not use and to report interval solutions. A mathematically more natural format should be possible. It's very difficult to convince students learning calculus to take the time to really understand and - it's hard enough to get them to understand intervals! (This might be a moot point as I do not find any references to in Maple 10.)