# Tagged Questions

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### How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
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### Proof of $\displaystyle\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$

I want to prove $$\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$$ without useing L'Hôpital's rule.
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### An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
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### Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
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### Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
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### How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx$$ in terms of the integral $$\int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
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### Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
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### Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
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### Difference between the two definitions about the equality of two functions

From a long time I have found there are two definitions about the equality of two functions (or identity of two functions). I quoted the two definitions in the following: Zorich's definition ...
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### Is there a differentiable function f which the differential function f' is bounded but has no maximum on a closed interval.

Is there a differentiable function $f$ in which the differential function $f'$ is bounded but has no maximum on one closed interval? Thanks
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### Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
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### Analysis question limit sin function as n goes to infinity

can you help me with the following: $\lim_{n \rightarrow \infty} \sin^{2} \pi \sqrt{n^2 + n}$ Thanks a lot!
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### Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question: Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists. I think this question have many example. But ...
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### sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
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### Taylor's Theorem Question

Assuming Taylor's Theorem holds: Given $$F(x)=f(b)-f(x)-\frac{f'(x)}{1!}(b-x)-\cdots-\frac{f^{(n-1)}(x)}{(n-1)!}(b-x)^{n-1}$$ Show that: $$F'(x)=\frac{-f^{(n)}(x)}{n!}(b-x)^{n-1}$$
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### Increasing, continuous, concave downward function normalised between 0.5 and 1

What would be a good function which is increasing, continuous, concave downward with $$\lim_{x \to 0} f(x) = 0.5$$ and $$\lim_{x \to \infty} f(x) = 1.$$ It should be concave downward whose ...
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### Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
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### nth term derivative with f(0) plugged in

Question: compute the sixth derivative of $\frac{\cos{(5x^2)}-1}{x^2}$ and plug in zero to the derivative. What is the answer? I keep concluding that this question should be 892857 when I plug in 0 ...