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Start with P(1+r/n)^nt. Let's call x=n/r. This means r/n = 1/x and n=xr.
So P(1+r/n)^nt = P(1+1/x)^xrt = P((1+1/x)^x)rt. The limit of (1+1/x)^x = e so P((1+1/x)^x)rt = Pe^rt.