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This article discusses a limiting behavior of breather solutions of thefocusing nonlinear Schr\"odinger (NLS) equation. These breathers belong to thefamilies of solitons on a non-vanishing and constant background, where thecontinuous-wave envelope serves as a pedestal. The rational Peregrine solitonacts as a limiting behavior of the other two breather solitons, i.e., theKuznetsov-Ma breather and Akhmediev soliton. Albeit with a phase shift, thelatter becomes a nonlinear extension of the homoclinic orbit waveformcorresponding to an unstable mode in the modulational instability phenomenon.All breathers are prototypes for rogue waves in nonlinear and dispersive media.We present a rigorous proof using the $\epsilon$-$\delta$ argument and show thecorresponding visualization for this limiting behavior.

Peregrine soliton as a limiting behavior of the Kuznetsov-Ma and Akhmediev breathers