Intereting Posts

density of sum of two uniform random variables $$
Is it always true that if $\gcd(a,b)=1$ then $\gcd(ab, c) = \gcd(a, c)\gcd(b, c)$?
Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Proof that $Γ'(1) = -γ$?
fair and unfair games
Ground plan of Backward direction (<=) – Let $p$ be an odd prime. Prove $x^{2} \equiv -1 \; (mod \, p)$ has a solution $\iff p\equiv 1 \; (mod 4)$
How to find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $?
Maximal area of a triangle
Linear independence of images by $A$ of vectors whose span trivially intersects $\ker(A)$
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
Prove equality in triangle inequality for complex numbers
Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?
Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.
What is the Möbius Function for graphs?
Infinite series of nth root of n factorial

Let $f:\Bbb R\to \Bbb R$ be a continuous function such that $\int _0^\infty f(x)\text{dx}$ exists.

Prove that if

$\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$

- If a function is undefined at a point, is it also discontinuous at that point?
- Discontinuous functions that are continuous on every line in $\bf R^2$
- Continuity of the function $x\mapsto d(x,A)$ on a metric space
- If $f$ has only removable discontinuities, show that $f$ can be adjusted to a continuous function
- Uniform continuous and not Hölder continuous
- A function $f:\to \mathbb R$ of bounded variation and absolutely continuous on $$ for all $\epsilon >0$ is absolutely continuous
If $f$ is non-negative then $\lim _{x\to \infty } f(x)$ must exist and $\lim_{x\to \infty} f(x)=0$

**My try**

To prove that $\lim_{x\to \infty} f(x)=0$ we should show that $\exists G>0$ such that $x>G\implies |f(x)|<\epsilon $ for any $\epsilon>0$

But I can’t find out how to show this.

Please help.

- Continuous Functions and Cauchy Sequences
- Evaluate the partial derivatives of the following function:
- A semicontinuous function discontinuous at an uncountable number of points?
- Continuity of absolute value of a function
- Are limits on exponents in moduli possible?
- Numerically estimate the limit of a function
- Limit problem arctg (1 ^ infinty)
- Does $\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt{n^{3}+1}\right)$ exist?
- Show that $p$ is prime if the following limit property holds
- $\lim_{{n}\to {\infty}}\frac{1^p+2^p+\cdots+n^p}{n^{p+1}}=?$

This is pretty standard. Let $F(x) = \int_{0}^{x}f(t)\,dt$ then we are given that $\lim_{x \to \infty}F(x) = L$ exists. Now by continuity of $f$ and fundamental theorem of calculus we have $F'(x) = f(x)$ and we are given that $\lim_{x \to \infty}f(x) = \lim_{x \to \infty}F'(x) = M$ also exists. We can see via mean value theorem that $$F(x + 1) – F(x) = F'(\xi) = f(\xi)$$ where $x < \xi < x + 1$. Letting $x \to \infty$ in the above equation and noting that $\xi \to \infty$ we get $$L – L = M$$ or $M = 0$ so that $f(x) \to 0$ as $x \to \infty$.

Second statement is false and counterexample is in the following figure (taken from the masterpiece *A Course of Pure Mathematics* by G. H. Hardy):

For all $n\in \mathbb{N}$ larger than 1,

Let

$$

f(x)=

\begin{cases}

0 \ for \ x\leq n

\\

n^4x-n^5 \ for \ n<x\leq n+1/n^3

\\

-n^4+2n+n^5 \ for \ n+1/n^3<x\leq n+2/n^3

\end{cases}$$

By some calculation

$\int_n^{n+1}f(x)dx=1/n^2$. Then $\int_0^\infty f(x) = \sum_{n=2}^\infty 1/n^2$.

But $lim_{x \to \infty} f(x)$ does not exist.

- Trace minimization when some matrix is unknown
- Could the concept of “finite free groups” be possible?
- Models of real numbers combined with Peano axioms
- Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$
- If p is prime and k is the smallest positive integer such that a^k=1(modp), then prove that k divides p-1
- How to compute the residue of a complex function with essential singularity
- Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$
- At large times, $\sin(\omega t)$ tends to zero?
- Asymptotic expansion of $\int_0^{2\pi}ae^{x\cos a}da$
- Expected number of frog jumps
- Is there a characteristic property of quotient maps for smooth maps?
- Why does the boundary of the Mandelbrot set contain a cardioid?
- Continuity of the Characteristic Function of a RV
- Two covering spaces covering each other are equivalent?
- Limits in the category of exact sequences