Recent content by a7d07c8114

Homework Statement I'm following along in Spivak's Calculus, 4th ed. In the chapter on complex functions (page 547 in my copy), Spivak outlines a proof that there is no "square root function" on the complex numbers; he leaves one step to the reader, claiming that it is a standard...

Ah. Yes, that does hold. So what the two of you are saying is that Spivak is finding \lim_{x\rightarrow a}\frac{F(x)}{G(x)} (for suitably defined F and G), and this equality is why the definition results in no loss of generality. That clears up everything! I wish Spivak had included this bit...

If the limits were any other number then maybe, but since the limits here are 0 we actually get continuity, a pretty big loss in generality if you ask me. (Don't actually ask me though, I'm clearly not the expert :P) Spivak's proof did basically what you did with p(x) and q(x), but I'm not sure...

Thanks to both for the replies, and thanks to Infinitum for the link. Infinitum: The proof for continuous differentiability in the link is pretty different from Spivak's proof. Spivak starts with a different set of hypotheses, namely that \lim_{x\rightarrow c}f'(x)/g'(x) exist and f and g be...

Homework Statement There is a part of Spivak's proof of L'Hospital's Rule that I don't really see justified. It is Theorem 11-9 on p.204 of Calculus, 4th edition. I won't reproduce the statement of L'Hospital's Rule here, except to say that it is stated for the case \lim_{x\rightarrow...

I haven't read it myself, but I've seen Axler's "Linear Algebra Done Right" recommended by quite a few posters when I searched for this same question.

Thanks to all those who replied. mathwonk, perhaps I put my friend's words in too negative a light, which I would never want to do. What he said, more precisely, was that it would be nice to see number theory in some form, but number theory is more or less optional before graduate school whereas...

A senior friend of mine who is going to graduate school in mathematics suggested that I try to get at least some exposure to number theory before applying to/attending graduate school. (I'm a freshman undergrad.) Well, I was going to do so anyway, since it's interesting and even applicable, but...

Thank you both for your replies. I have decided to pursue Spivak's Calculus on Manifolds first, followed by Calculus with Rudin to supplement if time permits. (I should have fun over summer break, and Calculus on Manifolds looks the most entertaining.)

I want to do some self-study both over the summer and during the school year, and I've chosen my books. They are: Rudin, Principles of Mathematical Analysis Spivak, Calculus + Calculus on Manifolds I will likely be taking a yearlong sequence starting next fall in analysis using Rudin, either...

Tried again the night after, seems you guys are right: it's gone. Appears to be lens flare. Thanks though!

Hi, I'm pretty new at this amateur astronomy thing, and I saw something through my telescope that I could not focus clearly enough on (as to be expected from a telescope I picked up from craigslist for $10). I would like help identifying it. Latitude: 40° 15' 36" N Longitude: 74° 16' 27" W...

Am I allowed to do that? It seems like just restating the definition of a recursive function to me...though I'm very new at this, so I don't know if that's what you're meant to do. Or maybe I'm just interpreting "uniquely defined" the wrong way.

Thanks for the reply. I'm not sure I understand very well though. Could you perhaps link me to an explanation or a proof that illustrates your point? I did feel that it was awkward writing the proof the way I did, but I'm so far a complete novice at proof writing in general.

Homework Statement Use the principle of mathematical induction to show that the value at each positive integer of a function defined recursively is uniquely determined. I understand the problem and its related concepts. However, I feel that my attempt at a proof doesn't use the principle...