# Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use (limits-colimits) instead.

3,333 questions
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### Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
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### A sufficient condition for a sequence to converge if arithmetic mean of the sequence converges?

We have a well-known conclusion: If a sequence $\{a_n\}_{n\in\mathbb{N}}$ converges, then the arithmetic mean $\frac{S_n}{n}$ (where $S_n=\sum\limits_{k=1}^na_k$ is the nth partial sum) converges to ...
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### Pythagoras tree bounding size

The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed ...
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### Assume $x_n>0$,$x_n+\dfrac{4}{x_{n+1}^2}<3.$ Prove $\lim\limits_{n \to \infty}x_n$ exists and evaluate it.

My Solution Notice that $$x_n+\dfrac{4}{x_{n+1}^2}<3=3\sqrt{\frac{x_n}{2}\cdot\frac{x_n}{2}\cdot\frac{4}{x_n^2}}\leq \frac{x_n}{2}+\frac{x_n}{2}+\frac{4}{x_n^2}=x_n+\frac{4}{x_n^2}.$$ This ...
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### Limit points of a sequence constructed from pi (if pi is normal)

I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind. If you make up a bunch of fractions of pi $3.141592\ldots$. If you start from the ...
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### Movement time of object with constant jerk, limited acceleration and velocity

A product is initially at rest on a conveyor belt: The initial conditions of the product can be described as follows:$$x_i=0$$ $$v_i=0$$ $$a_i=0$$$$j_i = j⋆$$. The product will be moved forward ...
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### A generalization of the Euler-Mascheroni constant

Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence $$f(n)-\sum_{k=1}^nf'(k).$$ If $f=\log$, then it is convergent to $-\gamma$ (...
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### A convergent sum of a divergent and convergent sequence?

Is the following argument correct? Sequences $(x_n)$ and $(y_n)$, where $(x_n)$ converges, $(y_n)$ diverges, and $(x_n+y_n)$ converges. Proof. The request in question is impossible. Assume that ...
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### Nonsmooth data in the conservation laws, their approximations and limits

In the book by R. LeVeque: "Numerical methods for conservation laws", Birkhauser, (1992), 2nd edition, in the Subsection 3.1.2 called "Nonsmooth data", the author talks about possibilities for finding ...
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### (Poisson limit theorem) Random variable $X_n$ ~ Bin$(n,p_n)$ convergences to $Z$ ~ Poisson($\lambda$)

$(\star)$ : Let $X_n$ ~ Bin($n,p_n$) ( $n \in \mathbb{N}$ ) $n \cdot p_n \rightarrow \lambda$ for $n \rightarrow \infty$. Then $X_n$ convergences to Poisson $Z$ ~ Poisson($\lambda$). I ...
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### Integral with Riemann Zeta Function. Trying to understand behavior

Hey Math Stack Exchange, I have an integral that I'd like to try and simplify or get bounds on. It involves the Riemann Zeta function which I'm not super familiar with and so I thought I'd look for ...
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### If $Z=X\cdot Y$ then $E[X^k]= \lim_{n \to k} \frac{ E[Z^n]}{ E[Y^n]}$ if $E[X^k]<\infty$

I am curious whether the following theorem is true. Let $Z=X \cdot Y$ where $X$ and $Y$ are two independent positive random variables. Moreover, suppose that $E[Z^k],E[Y^k]<\infty$ for all ...
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### showing that a multivariate function is not continuous at the origin, although Wolfram Alpha says it is

I am being asked to show that the following function is NOT continuous at the origin. \$f(x,y) =\begin{cases}\frac{x^4y^5}{x^8 + y^{10}} &&(x,y) \not= (0,0) \\0 &&(x,y) = (0,0) \end{...