If one wants to prove a statement for all values of a given parameter (e.g. ), then it often pays to first look at extreme cases when the parameter is very small (e.g. ) or very large (e.g. look at the asymptotic limit , holding other parameters fixed). If one can find a single method which works well for both extremes, then it is quite likely that it can also be extended to work for the intermediate cases as well. If instead one finds two different methods to deal with the two different extremes, then sometimes this indicates that one should divide into cases instead, for instance splitting into the case when when is less than some threshold, or when is above this threshold; this threshold can sometimes be left unspecified initially, and optimized later.
See also "look at small cases".