I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable calculus (mathematics 1). In mathematics 2 we're currently working on series (convergent, divergent, integral criteria, D'Alemberts criteria, Cauchy criteria, absolute convergence …). English is not my mother tongue, so forgive me I spell something wrong or have grammar mistakes. I'll try to explain my questions as best as I can. I have multiple questions, but they are all intertwined. Since all these things "need" limits, they are my main confusion.
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I understand the intuition behind the limit and the epsilon-delta definition, but why it works in practice. That is why can I say when computing the derivative of for example $x^2$ is $2x$? In $\lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) – f(x)}{\Delta x}$ I can't just put $0$ since I would get $\frac{0}{0}$, which would be the "true" derivative, because I don't know what that is. After some manipulation I would get $\lim_{\Delta x \rightarrow 0} 2x + \Delta x$ and since $\Delta x$ goes to $0$ that would be equal to $2x$. But this $\Delta x$ will never be $0$, at least as I look at this and from the definition of the limit it would say that I can make $\Delta x$ as close to $0$, but not equal to, if I'm willing to make $x_1$ and $x_2$ as close to each other. Why can I now take this $2x$ and say for instance that the derivative of someone's position it time is $2x$ that is its velocity is $2x$ and not $2x +$ some small $\Delta x$?
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When trying to see if an infinite series (which never ends) converges or diverges why can I look at a sequence of partial sums (infinite) of that series and based on their convergence or divergence say if the whole series diverges or converges?
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When I come to professors and ask these and such questions they tell me why am I bothering my self with such question and that I should take it for granted. Then I just want to kill my self. I mean haven't I came here to study how and why things work? I would like it more if they would just tell me that if it is some "higher" or more complex part of mathematics and that I will learn about it later or that it just isn't know why it works the way it works. So should I even continue to study these things, since I will always come across something that I wouldn't be able to understand (since these "basic" limits are confusing me) and all these professors and academia will tell me that I shouldn't worry why it works the way it works and that I should just take it for granted.
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All the theorems used to proof derivative, integral, convergence, divergence etc. use in one way or another limits. But in the definition of the limit it says that I can make some $f(x)$ as close to some value L, but not equal to it, as long as I'm willing to make $x$ as close to some value $c$. This definition is supposed to be mathematical rigorous, but using these as close don't "look" rigorous to me.
Please help me since I don't know should I even continue with my studies since there is always some mathematical proof which I cannot understand and is preventing me to go forward and that way I'm always lacking behind and everybody expects to understand everything the first time I hear it. I will be grateful for all comments and suggestions.
Best Answer
A function $f:\>x\mapsto f(x)$ given by some expression has a "natural" domain of definition $D(f)$: the set of all $x$ in the realm of discourse (${\mathbb R}$ or ${\mathbb C}$, say) for which $f(x)$ can be evaluated without asking questions. In most cases $f$ is continuous throughout $D(f)$, which means that for all $x_0\in D(f)$, when $x$ is sufficiently near $x_0$ then $f(x)$ is very near to $f(x_0)$.
Now some $f$'s may have "exceptional points" where they are not continuous, e.g., the sign-function, which is defined on all of ${\mathbb R}$, but is discontinuous at $0$. Above all, the set $D(f)$ may have "real" or "virtual" boundary points, where $f$ is a priori undefined. But nevertheless we have the feeling that $f$ has a "reasonable" behavior in the neighborhood of such a point. Examples are $x\mapsto{\sin x\over x}$ at $x=0$ (a "real" boundary point of $D(f)$), or $x\mapsto e^{-x}$ when $x\to\infty$ (here $\infty$ is a "virtual" boundary point of $D(f)$).
All in all the concept of "limit" is a tool to handle such "exceptional", or: "limiting", cases. An all-imortant example is of course the following: When $f$ is defined in a neighborhood of $x_0$ we are interested in the function $$m:\quad x\mapsto{f(x)-f(x_0)\over x-x_0}$$ which has an "exceptional" point at $x_0$. It is impossible to plug in $x:=x_0$ into the definition of $m$.
This brings me to your point 4. which gets to the heart of the matter. I'd rewrite the central sentence as follows: In the definition of the limit of $f(x)$ for $x\to c$ it says that I can make $f(x)$ as close to the value $L$ as I wish, as long as I'm willing to make $x$ sufficiently close to $c$. The idea is: While it is in most cases impossible to put $x:=c$ in the definition of $f$, we want to describe how $f$ behaves when $x$ is very close to $c$.
You then go on to say that "this definition is supposed to be mathematically rigorous, but using these as close and sufficiently close don't look rigorous to me".
The whole $\epsilon$-$\delta$ business serves exactly the purpose to make the colloquial handling of as close and sufficiently close that you are lamenting rigorous.
Life would be simpler if we could define $\lim_{x\to c}f(x)=L$ by the condition $|f(x)-L|\leq |x-c|$, or maybe $|f(x)-L|\leq 100|x-c|$. But four centuries of dealing with limits have taught us that the $\epsilon$-$\delta$ definition of limit, arrived at only around 1870 or so, captures our intuition about them in an optimal way. It takes care as well of the unforeseeable cases when the error $|f(x)-L|$ can be made as small as we we want, but we need an extra effort in the nearness of $x$ to $c$, e.g., $|x-c|<\epsilon^2$ instead of ${\epsilon\over100}$.