This is so wrong. The ratio between exp(x+ϵ) and exp(x) for even ϵ arbitrarily close to zero is infinity in the limit of large x. There is nothing like it with any polynomial growth for any exponent. The ratio between (x+ϵ)^p and x^p for arbitrarily large p is 1 in the limit of large x, for any ϵ.

Another sign that people don't get exponential growth, even if they think they do.