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 the tag “limits-colimits” instead.
45,108 questions
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
...
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Is this constructive method for determining the minimal ε–index of a convergent sequence already known?(Osamah banat theorem for determine least index [closed]
I am studying the ε–definition of convergence for real sequences.
Classical textbooks state that for every , there exists an integer such that
$$|a_n - L| < \varepsilon \quad \text{for all } n \ge ...
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3
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How to solve $ \lim\limits_{x\to+\infty} \!\!\left(\! \frac{x^{2}+3}{3x^{2}+1}\! \right)^{\!x^{2}}\!\!\!=0\;?$
I know it can be solved with Squeeze's theorem, but I want to verify that this more conventional method might also be valid.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 23
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What is $\lim_n \sqrt[n]{1+\cos(n)}$?
As I was going through some exercise list with limits, I found $\lim_n \sqrt[n]{1+\cos^2(n)}$. This is easy enough, since $\cos^2$ is bounded between 0 and 1, so a squeeze theorem argument lets us ...
- 13.2k
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Proving that the set of the quotients of Fibonacci numbers is not dense in the positive reals
As in the heading, I'm trying to write up a proof that the quotients of all Fibonacci numbers is not dense in $\mathbb{R_+}$. This is what I have come up with and would like to know if it's correct.
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Evaluate $\lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}$ [duplicate]
I am trying to evaluate the limit
$$
\lim_{n\to\infty}\left(\prod_{k=1}^n\left(1+\frac{k}{n}\right)\right)^{1/n}.
$$
My first thought was to take logarithms. That turns the product into a sum and ...
- 19
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6
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Find the limit $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
I have this limit
$$\lim_{n\to \infty} \sum_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$$ and and I tried to evaluate it in the following way:
First I set $x=\frac{k}{n^2}$ to make the expression ...
- 359
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Question about $\lim_{n\to\infty} \ln(n) =\infty$ [closed]
So I know that $\lim_{n\to\infty} \ln(n)=\infty$; I've seen some proof online using the mean value theorem. But is it not easier to assume that it converges, so that $\lim_{n\to\infty} \ln(n)=k$ where ...
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Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam.
Evaluate
$$
L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n
\left(
\left\lfloor\frac{2n}{k}\right\rfloor
-2\left\lfloor\frac{n}{k}\right\rfloor
\right),
$$
...
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Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
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Limit of the function satisfying $f(x)=x-f(x^2)$ as $x\to 1^-$
I guess $\lim\limits_{x\to 1^-} f(x) = 1/2$, where the function $f(x)$ defined by $f(x)=x-f(x^2)$ in $[0,1)$, or by the series:
$$
f(x) = x - x^2 + x^4 - x^8 + x^{16} - x^{32} + \cdots.
$$
I know $f(x)...
- 81
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1
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Proving a limit subject to an integral equation
Given a specific integral equation \begin{equation}
f(x) = 2- \frac{1}{\pi}\lim_{C\rightarrow \infty} \int^C_{-C}\frac{f(y)}{(x-y)^2+1} d y,
\end{equation}
I want to show that $$\lim_{C\rightarrow ...
- 855
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Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem
$$ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $$
My Work
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\cdot\frac{1}{3} \right)^{x^{2}\cdot\...
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1
answer
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Computing the third order directional derivative for a self-concordant function
Consider the concave function;
$$\phi(x) = \alpha+ \langle a, x \rangle -\frac{1}{2}\langle A x, x\rangle, \text{where} \;A=A^{T}\geq0, x,a \in \mathbb{R}^{n}, \text{and}\; \alpha \in \mathbb{R}$$
...
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Limit as $x\to 0$ of $Z(x)=\bigl[B(x I-A)C+D(x I+A)^{-1}E\big]^{-1} $
Let $A,B,C,D,E$ be $n\times n$ complex matrices. Assume that $B,C,D,E$ are invertible, and that $A$ is singular (non-invertible). Consider the matrix-valued function
\begin{equation}
Z(x)=\Bigl[B(xI-A)...
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