We show that if one drives the KPZ equation by the derivative of a space-time white noise smoothened out at scale ε≪1 and multiplied by ε3/4 then, as ε→0, solutions converge to the Cole-Hopf solutions to the KPZ equation driven by space-time white noise. In the same vein, we also show that if one drives an SDE by fractional Brownian motion with Hurst parameter H<1/4, smoothened out at scale ε≪1 and multiplied by ε1/4−H then, as ε→0, solutions converge to an SDE driven by white noise. The mechanism giving rise to both results is the same, but the proof techniques differ substantially.