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Why not include as a requirement that all functions must be continuous to be differentiable?

Because that suggests that there might be functions which are discontinuous at $a$ for which it is still true that the limit$$\lim_{t\to0}\frac{f(a+t)-f(a)}t$$exists. Besides, why add a condition ...
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Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?

Your only error -- and it's a common one -- is in a subtle misreading of L'Hopital's rule. What the rule says is that IF the limit of $f'$ over $g'$ exists then the limit of $f$ over $g$ also exists ...
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Why not include as a requirement that all functions must be continuous to be differentiable?

Definitions tend to be minimalistic, in the sense that they don't include unnecessary/redundant information that can be derived as a consequence. Same reason why, for example, an equilateral triangle ...
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

You are right to think of the probabilities as areas, but the set of points closer to the center is not a triangle. It's actually a weird shape with three curved edges, and the curves are parabolas. ...
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The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)

I'll try to give a soft answer to what I see as the spirit of the question, which is not why you get exactly log, but why the behaviour is different when integrating $x^{k}$ for $k=-1$. The way I see ...
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$100$-th derivative of the function $f(x)=e^{x}\cos(x)$

Find fewer order derivatives: \begin{align} f'(x)&=&e^x (\cos x -\sin x)&\longleftarrow&\\ f''(x)&=&e^x(\cos x -\sin x -\sin x -\cos x) \\ &=& -2e^x\sin x&\...
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Why is Euler's number $2.71828$ and not anything else?

$\sum\frac1{n!}$ is not that special. $\lim_{n\to\infty}\left(1+\frac1n\right)^n$ is not really special. $f'(x)=f(x)$ is a very simple differential equation, but unremarkable, really. $\ln (x)$ is ...
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Why can't the second fundamental theorem of calculus be proved in just two lines?

The problem with your proof is the assertion Now $dF$ is just the small change in $F$ and $f(x)dx$ represents the infinitesimal area bounded by the curve and the $x$ axis. That is indeed intuitively ...
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Motivation for the rigour of real analysis

In general, the push for rigor is usually in response to a failure to be able to demonstrate the kinds of results one wishes to. It's usually relatively easy to demonstrate that there exist objects ...
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Why and How do certain manipulations in indefinite integrals "just work"?

Many otherwise-mysterious tricks in integrals involving trigonometric functions can be explained by expressing the trig functions in terms of exponentials, as in $\cos(x)=(e^{ix}+e^{-ix})/2$. The ...
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Why can't calculus be done on the rational numbers?

This is a slightly softer answer. You can 'do calculus' in-so-far as you can define the derivative and perhaps compute some things. But you'll get no theorems out: the main interval theorems (the ...
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Intuition behind this interesting calculus result?

Yes, this was considered a pretty strange phenomenon when Torricelli first constructed such an example in 1643. (Torricelli himself found it so incredible that he offered two different proofs that the ...
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How can a "proper" function have a vertical slope?

No, we don't need two vertical points. By the same idea, if the graph of a function $f$ has an horizontal tangent line somewhere, then there has to be two points of the graph of $f$ with the same $y$ ...
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What does Big O actually tell you?

Reading between the lines, I think you may be misunderstanding Big O analysis as being a replacement for benchmarking. It's not. An engineer still needs to benchmark their code if they want to know ...
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Mathematical symbol for 'slightly greater than'?

More often it is used as $b=a+\epsilon$ where $\epsilon$ normally stands for a small positive quantity. That provides b slightly greater than a. Similarly $-\epsilon$ for slightly below.
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The $(\cdot)^{\frac{1}{4}}$ operation has to be understood as a function. A function can only have one image for any argument. Depending upon how you interpret the fourth root, the image could be ...