# Is there error in the answer to Spivak's Calculus, problem 5-3(iv)?

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.

• 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? Sep 18, 2011 at 0:31
• @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 $\delta_{1}$. Sep 18, 2011 at 3:14
• Are you sure the first $\delta$ that Spivak introduces is $\delta_2$? Seems a bit strange to me to name the first $\delta$ "$\delta_2$"... Sep 18, 2011 at 4:54
• 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.
– Did
Sep 18, 2011 at 9:30
• @Arturo: Yes, they are introduced on the opposite order on the key. Sep 18, 2011 at 14:12

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.