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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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Where you (correctly) iterated the bound twice it seems that Spivak iterated three times. This particular $\delta$ is shrinking at each iteration, because it satisfies $\delta(\epsilon,a) < \epsilon$ for all $a$. Given that two iterations are enough, three are more than needed, but still logically correct.

Without seeing the answer book, it is impossible to determine whether Spivak's extra layer of work is consistent with the methods he gives for this and other problems.