# Tagged Questions

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### Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$

Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$. How to prove that: The equation above has a unique solution $U_n$ ...
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### How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$

Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then $$\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0$$ My attempt: I ...
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### Existence of limit of $\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$!

If $f$ is continuous in $[0,1]$ then $$\lim_{n \to \infty}\sum_{i=0}^{[n/2]} \frac 1 n f \left(\frac i n \right)$$ (where $[y]$ is the largest integer less than or equal to $y$) (A) does not ...
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### lim of integration of a non-negative function.

i'd like very much your help with this one : given a positive function meaning $$f(x) \geq 0$$ and $f$ is continuous. Let $$M = \sup(f(x))$$ where $x$ belongs to $[a,b]$. How can i prove that ...
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### Two integration problems

Find out if this limit exists: $$\lim_{y\to 0}\int_0^1\frac{x e^{\frac{-x^2}{y^2}}}{y^2}dx$$ Evaluate $$\int_0^{\pi/2}\dfrac{\arctan(a \tan(x))}{\tan(x)}$$ I'm a bit lost with these two problems. I ...
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### Let $f>0$ differentiable in $[0,\infty)$. Assume $\lim \limits_{x \to \infty} (\log\circ f)^\prime(x) < 0$. Show that $\int_0^\infty f$ converges.

So what I gathered from the givens about $f$, since $(\log\circ f)^\prime(x)=\frac{f^\prime(x)}{f(x)}$ it would mean that far enough, $f^\prime(x)<0$. I don't know how to go about this from here. ...
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### When does $\lim\limits_{n\to\infty}\int_{b}^{a_n}f_n(x)dx=\lim\limits_{n\to\infty}\int_b^\infty f_n(x)dx$ hold?

Let $\{a_n\}\subset \mathbb{R}$ be sequence and $$f_n:[b,\infty)\longrightarrow \mathbb{R}, \qquad n=1,2,\dots .$$ Assume that $$\lim_{n\longrightarrow\infty}a_n=+\infty.$$ Obviously, from the ...
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### Symmetric limits

I read that you can use something called a "symmetric limit" to evaluate the improper integral of sin(x) as 0, something that I attempted by taking the right hand improper integral and adding it to ...
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### Lebesgue integrable function and limit

Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$. My solution which is ...
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### Evaluating $\lim \limits_{n\to \infty} \left( n \int_{0}^{\frac \pi 2} 1-\sqrt [n]{\sin x} \,\mathrm dx \right)$

Evaluate the following limit: $$\lim \limits_{n\to \infty} \;\; n \int_{0}^{\frac \pi 2} \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx$$ I have done the problem . How I solved is First I ...
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### Prove the existence of a limit : $\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}=0$

Let $F(x),G(x)$ be nonnegative decreasing functions in $[0,+\infty)$, with$\,\displaystyle \lim_{x\rightarrow+\infty}{x(F(x)+G(x))}=0$ (1) Prove that: $\forall \varepsilon>0$,we have ...
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### $f$ continous at $x_0$ $⇒\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$

The function $f:ℝ→ℝ$ is continuous on $x_0\inℝ$. Prove using the definition of a Darboux Integral that $$\lim_{h→0}∫_{x_0}^{x_0+h}\frac{f(t)}{h}=f(x_0)$$ I'm a first grade math student following an ...