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.
45,104 questions
0 votes
0 answers
29 views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam. Evaluate $$ L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \left( \left\lfloor\frac{2n}{k}\right\rfloor -2\left\lfloor\frac{n}{k}\right\rfloor \right), $$ ...
0 votes
0 answers
76 views
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
-4 votes
4 answers
227 views
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 $$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\cdot\frac{1}{3} \right)^{x^{2}\cdot\...
5 votes
0 answers
86 views
Limit as $x\to 0$ of $Z(x)=\bigl[B(x I-A)C+D(x I+A)^{-1}E\big]^{-1} $
Let $A,B,C,D,E$ be $n\times n$ complex matrices. Assume that $B,C,D,E$ are invertible, and that $A$ is singular (non-invertible). Consider the matrix-valued function \begin{equation} Z(x)=\Bigl[B(xI-A)...
- 597
1 vote
1 answer
76 views
Can $\lim\limits_{(x,y)\to0}f(x)$ doesn't exist but at every path $y=\sum\limits_{k=1}^na_k x^k$, $\lim\limits_{(x,y)\to0}f(x)$ exist and are equal.
This is a generalization of this question A quick and easy was to prove that a 2 dimensional limit like $$\lim\limits_{(x,y)\to0}\frac{xy}{x^2+y^2}$$ is to try 2 different linear paths and prove that ...
- 9,055
1 vote
0 answers
61 views
Motivation for constructing auxiliary functions in a proof that $f(x) \to0$ given a differential inequality
I'm working on a problem in analysis and I understand the steps of the proof for one of its cases, but I'm struggling to understand the motivation behind the specific construction used. I'd appreciate ...
- 11
6 votes
1 answer
131 views
Finding $\lim_{n \to \infty} \frac{l(n)}{n^2}$ for minimal vertex coloring satisfying a property for all $k \times k$ sub-squares
I am working on the following grid coloring problem and am stuck on finding the general form of $l(n)$. The Problem Some of the vertices of the unit squares of an $n \times n$ chessboard are colored ...
- 681
0 votes
2 answers
147 views
Determine the convergence or divergence of $\sum_{n=1}^\infty\frac{1}{n^{\arctan n}}$
There was a problem in my textbook: Determine the convergence or divergence of $$ \sum_{n=1}^\infty\frac{1}{n^{\arctan n}} $$ The method my instructor taught me is that, notice that this sum is ...
- 143