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I'm puzzled by the answer to a problem for Spivak's Calculus (4E) provided in his Combined Answer Book.

Problem 5-3(iv) (p. 108) asks the reader to prove that $\mathop{\lim}\limits_{x \to a} x^{4} =a^{4}$ (for arbitrary $a$) by using some techniques in the text to find a $\delta$ such that $\lvert x^{4} - a^{4} \rvert<\varepsilon$ for all $x$ satisfying $0<\lvert x-a\rvert<\delta$.

The answer book begins (p. 67) by using one of these techniques (p. 93) to show that $$\lvert x^{4} - a^{4} \rvert = \lvert (x^{2})^{2} - (a^{2})^{2} \rvert<\varepsilon$$ for $$\lvert x^{2} - a^{2} \rvert <\min \left({\frac{\varepsilon}{2\lvert a^{2}\rvert+1},1}\right) = \delta_{2} .$$

In my answer, I use the same approach to show that $$\lvert x^{2} - a^{2} \rvert <\delta_{2}$$ for $$\lvert x - a \rvert <\min \left({\frac{\delta_{2}}{2\lvert a\rvert+1},1}\right) = \delta_{1} ,$$ so that $$\lvert x^{4} - a^{4} \rvert<\varepsilon$$ when $$\delta = \delta_{1}=\min \left({\frac{\delta_{2}}{2\lvert a\rvert+1},1}\right). \Box$$

But Spivak's answer book has $$\delta =\min \left({\frac{\delta_{1}}{2\lvert a\rvert+1},1}\right),$$ which I believe is an error.

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    If it is incorrect, perhaps you can find a particular value of $a$ and a particular value of $\epsilon$ where his formula fails. Can you?2011-09-18
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    @GEdgar: That may be worth determining, but the question at hand is, first, should it be obvious that $\delta_{1}$ was intended, and if so, what step am I missing, since applying the techniques of the chapter, as well as all the steps explicitly worked out in the answer key, leads to $\delta_{2}$ where it ends up with $\delta_{1}$.2011-09-18
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    Are you sure the first $\delta$ that Spivak introduces is $\delta_2$? Seems a bit strange to me to name the *first* $\delta$ "$\delta_2$"...2011-09-18
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    Since $\delta_2<\varepsilon$, $\delta_1<\delta_2$ and Spivak's $\delta$ (based on $\delta_1$) is smaller than yours (based on $\delta_2$). Thus, if your $\delta$ is correct, Spivak's $\delta$ is correct as well. So much for counterexamples.2011-09-18
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    @Arturo: Yes, they are introduced on the opposite order on the key.2011-09-18
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    @Didier: Yes; but the logic of the chapter, and particularly of this exercise, is the the make the step from the bounds on $\lvert x^{4} - a^{4} \rvert$ to the bounds on $\lvert x - a \rvert$ through the intermediate step of the bounds on $\lvert x^{2} - a^{2} \rvert$, and to make the *corresponding substitution* (here $\delta_{2}$ where Spivak has $\delta_{1}$). To make a *different* substitution (without even noting that $\delta_{1} \le \delta_{2}$) will certainly leave readers wondering (especially since, as he doesn't perform further calculations, there's no motivation for doing so.)2011-09-18
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    That may be so. I simply mentioned a logical implication which made moot the task of looking for *counterexamples* and which some of the people interested by your question did not notice, apparently.2011-09-18
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    @Didier: Yes, understood. Though your point is a good one and I should perhaps add the remarks in my comment above to the question to clarify that the issue is not whether the key's answer _works_.2011-09-18
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    Where does $\delta_1$ first show up in Spivak's answer?2011-09-19
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    @MichaelChen: Right at the top. It shows up first because it's something that's been shown in the text, so he mentions it first.2011-09-19

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