In Chapter 5, Problem 41, Spivak provides an alternative way to prove that
$$\lim_{x \rightarrow a} x^2 = a^2\,\,,\,\,a > 0$$
Given $\,\epsilon > 0\,$ let
$$\delta = \min\left\{\sqrt{a^2 + \epsilon} - a, a - \sqrt{a^2 - \epsilon}\right\}$$
Then
$$|x - a| < \delta\Longrightarrow \sqrt{a^2 - \epsilon} < x < \sqrt{a^2 + \epsilon}\Longrightarrow a^2 - \epsilon < x^2 < a^2 + \epsilon\,\,,\, |x^2 - a^2| < \epsilon$$
Then he goes on to claim that this proof is fallacious. But wherein lies the fallacy?