On small homotopies of loops

Two natural questions are answered in the negative:
• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”
• “Can adding arcs to a space cause an essential curve to become nulhomotopic?” The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π1-shape injective.