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The limit $\displaystyle \lim_{x \rightarrow 0 } \frac{x}{x}=1$, as you said. It is not infinity. Why, because the fraction is simplified to $1$. So no matter where $x$ tends the limit will always be $1$. As for undefined it means that something is not defined. For example the function $f(x)=\frac{1}{x}$ is not defined at $x_0=0$, that is it is undefined at ...

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We like to think about continuous functions. Everything to us is continuous, and the basics of physics and calculus, where limits came from, dealt mainly with continuous functions. Why does that matter? Because a continuous function is exactly a function $f$ such that $f(x_0)=\lim_{x\to x_0}f(x)$. So when we want to calculate, for example, $\lim_{x\to ... 3 One possibility is to look what gets published at arXiv.org. Skim the abstracts to get an overview and if something interests you, you can find more on that topic. 3 Yes, there is a way to profitably read mathematical proofs, but it takes time. Here is an excerpt from the "note to the reader" in an excellent topology book: "It is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it ... 3 It is hard to get a good overview of all of mathematics. The best really is to take a good broad variety of classes. This you usually do (to some extent) the first couple of years in graduate school. Here you will learn the basic language of the main areas of mathematics. Here you will be get to meet various professors. For now, then, I wouldn't worry too ... 3 I think it's important to know first how deeply you want to study differential geometry/differentiable manifolds. I agree completely with Mike Miller's comment above, but would like to add a few thoughts. There is a big distinction between just studying differentiable manifolds and differential geometry. Geometry relies entirely on the definition of ... 3 The symbols$\gg$and$\ll$don't have a formal definition. Usually they are used to compare two extremely big numbers, for example$\mathrm{Graham's \; number} \ll \mathrm{TREE}(3)$or something like that. They are only used because someone wants to make clear that one of them is so much greater. The numbers are indeed just small compared to everyday ... 2 what is the Significance / usefulness of a set being closed? To this very broad question, I would say that, in my opinion, in many instances where I have seen that establishing the closure of a set is important is because it is a necessary condition for compactness in complete normed spaces. what more can be deduced when a set is proved to be closed ... 2 As you finish your undergraduate studies, you should have had at least some introduction or passing acquaintance to the following subjects: Algebra (the abstract kind) Discrete mathematics (maybe some number theory) Linear algebra Real/complex analysis (post-calculus) Topology Differential equations Statistics/probability Of these, which did you ... 2 An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998): Conjecture: The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4. See for example Bounded - Yes, but 4? and references therein. 2 I think that part of this question answers itself when you get "into" a field. When I was writing my MSc. dissertation I really had no idea what I was interested in, but my supervisor pointed me in a few directions and I ended up working in the one I found most interesting. Then, as you read more and more papers and see the most recent research you find ... 1 Here are some advantages of Sage (also known as SageMath) relevant to your question. Sage supports all fields of mathematics, including those you mentioned (groups, finite fields, combinatorics, permutation groups, linear algebra over finite fields, linear programming, integer programming). Sage relies on Python. it is a widespread, well-designed ... 1 This answer is something of a counterpoint to Samuel's enthusiastic answer. In short (and to directly answer your question), I would say that you are much better off just using Python (or other constituent programs) directly. Exactly which of these you might use depends very much on what you want to do. While Sage relies on Python, it is not a simple Python ... 1 If you are in the U.S., you could try attending AMS Sectional Meetings. These sorts of conferences happen often, and they feature talks in a huge number of different topics in current mathematics. Otherwise, I'm sure that in your locale, there are probably analogous conferences. Also, you could sign up for email notifications from arXiv mathematics. You ... 1 This is a problem of "reality", not mathematics. If you have$n$different, indistinguishable objects and you pick one at random, than the probabilty for every object to be picked is$p=\frac{1}{n}$. However, you're problem arises, when you assigning the picking process to a person. A person might not pick fully at random. An observation from a magicians ... 1 I agree with programming/computation, probability/statistics, and linear algebra from Comments. Also, Optimization from Answer. Would add group theory. Find out what computer languages are currently in greatest use as the time graduation gets near, especially for managing and parsing large datasets; learn the basics of all. Unless the landscape changes ... 1 Big Data is a very broad definition. If want to work in data-mining or machine learning my list would start with these Statistics/Measure theory Optimisation (in general and especially convex optimisation) Funcional Analysis 1 The received wisdom is that pretty well any mathematical statement can in principle be formulated and hence formalised in ZF. I think this is rather an overstatement, but let's bear with it. So in some sense, logic can be viewed as providing a formal foundation for mathematics. However, to do logic rigorously, you need to be able to define and reason about ... 1 From my understanding, you can use set theory to model the objects studied in mathematics. Then major mathematical theorems can be proven from these models. Take for example, ZFC. If you accept the axioms, then you can use the language of first-order logic to deduce theorems. In Enderton's book on set theory (Elements of Set Theory), he gives a very easy to ... 1 Not the best answer, but I'll try with a short answer. Some of the set theory books, you will find it says for example$A\subseteq A\cup B$, where$A$and$B$are both sets. If we want to check if it's true, we have to prove that the statement$(x\in A)\implies (x\in A\cup B)$is true. Note that this is equivalent to$(x\in A)\implies [(x\in A)\vee (x\in ...

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Try looking at Vilenkin's introductory 1968 book on Representation Theory and Special Functions. Then look at his 3 volumes on the same subject with a more detailed treatment.

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Have you tried reading "A sequel to the first six books of Euclid" by Casey? There are other books by Casey on Plane Trigonometry and Spherical Trigonometry. There is also a book on geometric conic sections by William Wallace and another smaller one on geometric conic sections by by W.H. Besant with problems and solutions. Then there is "Constructive ...

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Derivatives on fractals sets was considered by several authors: Jiang, H. & Su, W. Some fundamental results of calculus on fractal sets Communications in Nonlinear Science and Numerical Simulation , 1998, 3, 22 - 26; Parvate, A. & Gangal, A. Fractal differential equations and fractal-time dynamical systems Pramana J Phys, Springer India, 2005, 64, ...

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