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
-1 votes
0 answers
74 views
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 ...
2 votes
1 answer
121 views
Limit with floor sums reminiscent of the exponent of the central binomial coefficient
This problem comes from the 1976 Putnam exam. Evaluate $$ L=\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \left( \left\lfloor\frac{2n}{k}\right\rfloor -2\left\lfloor\frac{n}{k}\right\rfloor \right), $$ ...
- 23
-3 votes
0 answers
47 views
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??
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
0 votes
1 answer
55 views
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 $$\lim_{C\rightarrow ...
- 855
-4 votes
4 answers
230 views
Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem $$ \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\...
0 votes
0 answers
31 views
Computing the third order directional derivative for a self-concordant function
Consider the concave function; $$\phi(x) = \alpha+ \langle a, x \rangle -\frac{1}{2}\langle A x, x\rangle, \text{where} \;A=A^{T}\geq0, x,a \in \mathbb{R}^{n}, \text{and}\; \alpha \in \mathbb{R}$$ ...
- 123
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 votes
0 answers
31 views
Find the limit $ \lim_{n \to \infty} \frac{\int_0^1 f^n(x) \ln(x + 2) \, dx}{\int_0^1 f^n(x) \, dx} $ [closed]
Let $( f(x) = -x^3 + 2x^2 - x + 1 $. Find the limit $$ \lim_{n \to \infty} \frac{\int_0^1 f^n(x) \ln(x + 2) \, dx}{\int_0^1 f^n(x) \, dx} $$ Note on the Solution Approach It seems that the Mean Value ...
1 vote
1 answer
78 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 ...
- 9,127
0 votes
1 answer
84 views
Can $\lim\limits_{(x,y)\to0}f(x)$ doesn't exist but at every path $y=mx$, $\lim\limits_{(x,y)\to0}f(x)$ exist and are equal. [duplicate]
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 they aren't equal or that the limit ...
- 9,127
1 vote
0 answers
62 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
0 votes
1 answer
98 views
Got stalled with a simple sequence problem: $x_1 = a$, $0<a<1$; $x_{n+1} = \ln(1+x_n)$; $\lim_{n\to \infty} nx_n$; [closed]
Define $\{x_n\}$ sequence this way: $$x_1 = a, \quad 0<a<1 $$ $$x_{n+1}=\ln(1+x_n)$$ $$\lim_{n\to \infty}nx_n \rightarrow \;?$$ While it’s not hard to prove $\lim_{n\to \infty}x_n=0$, I still ...
6 votes
1 answer
132 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 ...
- 833
-1 votes
3 answers
118 views
Find $ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $
$$ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $$ I know that I am not making any change in the expression, I am just re-expressing it $$ \lim_{x\to \infty} \left( 1+\frac{-5}{x+1} \...