Eguchi-Hanson metric from various limits

In this note, we review various seemingly different ways of obtaining Eguchi-Hanson metric with or without a cosmological constant term. Interestingly, the conformal class of metric corresponding to hyperbolic $n$-monopole solution obtained from the generalized Gibbons-Hawking ansatz, reduces to the Eguchi-Hanson metric in a particular limit. These results, though known from an algebraic geometry point of view, are useful while dealing with rotational killing symmetry of self-dual metrics in general theory of relativity as well as in the context of duality symmetry in string theory.