Question: What goes wrong with this example even though the Euler's method is shown to be stable. This points to the shortcomings of the definition of stability given above. These are

•  the definition is applicable in the limit of small step size, and

•  the definition allows some growth of the solution for bounded times.

Remark: If the solution of the IVP is stable or asymptotically stable, we cannot tolerate any growth in the computed solution and if the solution of the IVP is unstable, some growth of the perturbations is acceptable. The following concept of absolute stability will provide a more useful tool when the solutions of IVP are not growing in time.