integral{from t=0, to t=x} t^{-1+0} dx but you calculate instead f(x) = lim_{ε->0} [ integral{from t=0, to t=x} t^{-1+ε} dx ] that is just = lim_{ε->0} [ (t

integral{from t=0, to t=x} t^{-1+0} dx but you calculate instead f(x) = lim_{ε->0} [ integral{from t=0, to t=x} t^{-1+ε} dx ] that is just = lim_{ε->0} [ (t^ε - 1) / ε ] replacing t^ε = lim_{ε->0} [ (e^(ln(t)ε) - 1) / ε ] by https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule = lim_{ε->0} [ ln(t)e^(ln(t)ε) / 1 ] that is easy to calculate = ln(t)e^(ln(t)0) / 1 = ln(t)1 / 1 = ln(t) So f(t)=ln(t) reply