# 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.

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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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### Compute $\lim\limits_{n\to\infty}(x_{n+1}-x_n)$ if $x_n =\sum\limits_{k=1}^{n-1}f(\frac kn)$ and $f$ continuous (but not continuously differentiable)

The following question from Furdui's book (Exercise 1.32. page 6) is an "open problem" : Let $f: [0,1] \to \mathbb{R}$ be a continuous (and not a continuously differentiable) function and ...
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### Evaluate $\lim\limits_{x\to 0 } \frac{ e^{\frac{\ln(1+ax)}{x}} - e^{\frac{a\ln(1+x)}{x}}}{x}$.

Problem Evaluate $$\lim_{x\to 0 } \frac{ e^{\frac{\ln(1+ax)}{x}} - e^{\frac{a\ln(1+x)}{x}}}{x}.$$ Solution For convenience, denote $u(x)=\dfrac{\ln(1+ax)}{x}$ and $v(x)=\dfrac{a\ln(1+x)}{x}.$ ...
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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 shows ...
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### Fraction of $1$s in binary representation of $n!$

I plotted a fraction of $1$s in binary representation of $n!$ (i.e. A079584/A072831) for $n$ from $1$ to $10^4$: It appears it might converge to some limit for $n\to\infty$. Can we (dis-)prove that ...
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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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### Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
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### A Family of Limits Leading to an Interesting Function

A while back I got very interested in limits of the form $$\lim_{n\to\infty} (2A)^n \left (A-\underbrace{\sqrt{a+\sqrt{a+\ldots\sqrt{a+z}}}}_{n\textrm{ radicals}} \right )=f_a^{-1}(z)$$ Where $A$ ...
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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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### 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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### Oscillating integral converges to zero?

An integral like, say, $$\int_0^1 \cos[ nf(x)]~dx$$ with some function $f$ which is well behaved, and maybe almost everywhere non-zero, should be very small for large $n$ since the positive and ...
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### $\lim\limits_{n\rightarrow +\infty}\int_0^{2k\pi}\sqrt{\sin^{2n}(x)+\cos^{2n}(x)}dx$

I think the limit value of the function $\sin^{2n}(x) + \cos^{2n}(x)$ when $n$ tends to infinity is $0$ for all real values of $x$ except when $x$ is an integral multiple of $\pi$, where it comes out ...
### Proof verification: $\lim\limits_{x\to\infty}f(x)=L\Leftrightarrow \lim\limits_{x\to 0^{+}}f\left(\frac{1}{x}\right)=L$
Problem 81(a) of James Stewart's Calculus: Early Transcendentals 8e asks us to prove that $\lim\limits_{x\to\infty}f(x)=\lim\limits_{x\to 0^{+}}f\left(\frac{1}{x}\right)$, provided that these limits ...
### If $(a_n)$ is a sequence such that $a_n=a_{f(n)}+a_{g(n)}$, where $\lim \frac{f(n)}{n}+\lim\frac{g(n)}{n}<1$, can we claim that $\lim\frac{a_n}{n}=0$?
The inspiration for this question came with an attempt to solve this. Let $(a_n)_{n\in\Bbb{N}}$ be a sequence of real numbers satisfying $a_n = a_{f(n)} + a_{g(n)}~\forall n\in\Bbb{N}$, where \$f, g: \...