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.
44,881 questions
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Why do I need the extra manipulation when calculating a limit? [closed]
When calculating $\lim _{x\to \:\infty \:}x\left(\sqrt{x^2+1}-x\right)$, I didn't see the need to multiply it by the conjugate of $\sqrt{x^2+1}-x$, which got me zero as a result. When checking my ...
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Evaluating of Multidimensional Integral
\begin{equation}
\lim_{n \to \infty}\int_0^1\int_0^1\dots \frac{1}{1 - \frac{\ln(x_1x_2\dots x_n)}{n}}\,\mathrm dx_1 \mathrm dx_2 \dots \mathrm dx_n
\end{equation}
I've want to try apply a ...
- 381
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Taking a variant of the "birthday problem" to the limit as it approaches infinity
The birthday problem, in it's basic form, asks what is the chance that there will be a shared birthday in a group of $n$ people. There are also many variants, where one can see the linked paper ...
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Prove for value at risk $V_\alpha(-L)=-V_{1-\alpha}(L)$
Let $V_\alpha(L)$ be the value at risk of a function $L$ at $\alpha\in[0,1]$,
$$V_\alpha(L)=\inf\{x:P(L\le x)\ge\alpha\}.$$
Prove that for any $\alpha$,
$$V_\alpha(-L) =-\lim_{\epsilon\rightarrow0^+}...
- 2,356
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How many different "types" of zero are there? [closed]
Apologies if this may come off as a stupid question, but my friends and I (currently taking A-level further mathematics) were discussing the probabilities for individual points on a normal ...
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2
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Prove that $\lim_{n\to\infty} n\left(1- \frac{\ln n + c} n\right)^{n-1} = e^{-c}$
I am trying to show that $\lim\limits_{n \rightarrow \infty} \left( n \left(1- \dfrac {\ln n + c} n\right)^{n-1} \right) = e^{-c}$ in order to solve a certain H.W. problem, where I am confident that ...
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Question about a proof of Fermat's theorem/interior extremum theorem
My question is about the proof on the wiki. It specifies that $x>x_0$, meaning that $x-x_0>0$, and since $f(x_0)$ is a maximum by assumption, then it seems to me that $f(x)-f(x_0)$ should be ...
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Proof that the degree of a function doesn't change if adding to a constant if the degree itself is positive.
Note: several comments note that the degree of the function needs to be positive for this formula to hold.
Given a function $f$, we define its degree by the following limit:
$$
\deg f(x) = \lim_{x \...
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Can two functions be equivalent at a point and at the same time their limit do not exist?
I have a question about the use of ∼, the asymptotic equivalence.
Can we use it between two functions at a point, knowing at the same time that the limit of any of ...
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How could one guess that $\lim_{n\to\infty}\int_E f(nx)dx = \mu(E)\int_0^1 f(x)dx$? [duplicate]
Problem $1$ from UCI's Fall 2015 Real Analysis Qualifying Exam asks us to calculate
$$\lim_{n\to\infty}\int_E f(nx)dx$$
given that $f\in C(\mathbb R)$ is $1$-periodic and $E\subseteq [0,2\pi]$ is ...
- 6,369
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Valid exchange of limit and integral operations?
Is it possible to integrate a function that requires limit-taking? For example, set
$$f(x) := \frac{1}{2^{2 a} x^{2 a}+1}$$
$\underset{a\to \infty }{\text{lim}}f(x)$ is a square impulse of amplitude ...
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Evaluation of the Limit $\lim_{n\to\infty}\frac{n+n^2+n^3+\cdots +n^n}{1^n+2^n+3^n+\cdots+ n^n}$ with Rigorous Bounds
The following limit from the JEE pops up around the internet from time to time:
$$\lim_{n\to\infty}\frac{n+n^2+n^3+\cdots+ n^n}{1^n+2^n+3^n+\cdots +n^n}$$
While I agree that the first few steps are ...
- 336
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Approximating the Exponential Function Using Functions Asymptotically Insensitive to Scale
I am looking for a concrete formula for a sequence $f_n : \mathbb{R} \to \mathbb{R}$ of functions, as simple as possible, such that :
$f_n(x) \to e^x$ as $n \to \infty$ holds for every $x \in \mathbb{...
- 3,156
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Evaluation of the Limit of the Sum without using integral [closed]
I am trying to evaluate the following limit:
$$
\lim_{n \to \infty} \sum_{i=1}^{n} \frac{\sqrt{1 + \frac{i^2}{n^2}}}{n}
$$
I know this sum can be evaluated using integrals, but I’m curious if there’s ...
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$\frac{\pi-P_i}{P_c-P_i}$, for $P_i$ and $P_c$ the perimeters of in- and circum-scribed $n$-gons of a circle of radius $1/2$, as $n$ goes to infinity
Archimedes' method for calculating uses inscribed and circumscribed n-gons to bound the value for $\pi$. Now for a circle with radius 1/2, let $P_i$ be the length of the perimeter of an inscribed n-...
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