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.

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Follow-up: $h_{\omega}(s)$'s so that $\lim_{\omega\rightarrow 2\pi n} \frac{e^{-i\omega s} - 1}{e^{-i\omega} - 1} + h_{\omega}(s) = e^{-i2\pi n t}s$

This is a follow-up question or variant question to Finding $1$-periodic $h_{\omega}(s)$'s so that $\lim_{\omega\rightarrow 2\pi n} \frac{e^{-i\omega s} - 1}{e^{-i\omega} - 1} + h_{\omega}(s) = s$....
Maximal Ideal's user avatar
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21 views

The Raabe test limit existence implies the sequence goes to $0$

Let $b_n$ be a sequence of positive real numbers such that the limit from the Raabe-Duhamel's test is positive, i.e. $$\lim_{n \to \infty} n\left( \frac{b_n}{b_{n+1}}-1\right)=p > 0$$ the limit ...
Shthephathord23's user avatar
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1 answer
39 views

I want to show that $sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n.x^{2n+1}}{(2n+1)!}$ by showing $R_n(x)\rightarrow0$ when $n\rightarrow\infty$

I tried something like that: For $R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{n+1}}{(n+1)!}$ when tried to take the limit for $c=0$ I get; $lim_{n\rightarrow\infty}\frac{(2n+1)(x^{2n})(x^{n+1})}{(2n+1!)}$ (...
Elfryionnn's user avatar
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0 answers
16 views

Partial limits, question clarification.

I don't understand a question and I need clarification as to what is being asked. The question is: Assume that $\forall n, \frac{n}{n+1}$ is a partial limit of the sequence $\left\{{a_n}\right\}^\...
Someguyalive's user avatar
1 vote
1 answer
23 views

Finding $1$-periodic $h_{\omega}(s)$'s so that $\lim_{\omega\rightarrow 2\pi n} \frac{e^{-i\omega s} - 1}{e^{-i\omega} - 1} + h_{\omega}(s) = s$

One can show that the expression $$ \frac{e^{-i\omega s} - 1}{e^{-i\omega} - 1} $$ approaches $s$ as $\omega\rightarrow 0$ by L'Hopital's rule. However, it seems there is no convergence when ...
Maximal Ideal's user avatar
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60 views

$\lim\limits_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$

$\lim\limits_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$ obviously L'hopital is inapplicable here, I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is ...
Qusai Saify's user avatar
1 vote
0 answers
33 views

Proof clarification

The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is : Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be ...
Someguyalive's user avatar
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1 answer
24 views

limit of convergane in probability

I was learning about the convergence in probability. I'm unsure if looking at the epsiolon-delta definitions below which captures better the convergence in probability correctly. $$ \forall \epsilon,\...
Tomer Gigi's user avatar
1 vote
1 answer
41 views

Proof of L'Hospital's Rule, Part 2 - Question about inequality

I was watching this video https://www.youtube.com/watch?v=cHvYlrud8j8&ab_channel=AndrewMcCrady and in the proof, he states that when $0 \lt \delta = \min (1, \epsilon, \frac{\epsilon}{|L| + 1}) $, ...
ToonyDays's user avatar
1 vote
0 answers
45 views

Definition of limit involving open sets

I recently had a bit of an obsession with the definition of limits, and that made think about this function: $$f: \mathbb{Q} \to \mathbb{R}; \ \ \ \ f(x) = x$$ What is $\ \lim_{x \to 0} f(x)\ $? ...
the thinker's user avatar
1 vote
1 answer
44 views

Need help with this differential equation problem involving second derivative and first derivative

I have been trying to solve this problem. I was able to observe that from the given conditions $x^2 + f^2(x) + (f'(x))^2 = c$, where c is some constant, but it does not help me in any of the options. ...
zynox's user avatar
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Limit of the sequences $\left(\frac{1}{2^n}\binom{n+k}{k}\right)$ and $\left(\sqrt[n]{n^3}\right)$? [closed]

I need to show that $$\left(\frac{1}{2^n}\binom{n+k}{k}\right)_{n\in \mathbb{N}}$$ for a fixed $k \in \mathbb{N}$ and $$(\sqrt[n]{n^3})_{n\in \mathbb{N}}$$ converge, and find their limit. For the ...
QuickSilver1996's user avatar
1 vote
1 answer
54 views

Let $(x_n)_{n\in\mathbb N}$ be a real sequence in $[0,1]$. Is there a partition of $\mathbb N$ indexed by the limit points of $(x_n)$?

Let $(x_n)_{n\in \mathbb N}$ be a sequence of reals in $[0,1]$ indexed on the positive integers. Let $L\subset [0,1]$ be the limit points of $(x_n)$, in the sense that $l\in L$ if and only if there ...
jlewk's user avatar
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0 answers
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Proof of L'Hospital's Rule, Part 2

So I was watching this video https://www.youtube.com/watch?v=cHvYlrud8j8&ab_channel=AndrewMcCrady and along the proof it says that $(L-\epsilon)(1-\frac{g(c)}{g(\alpha)}) + \frac{f(c)}{g(\alpha)} \...
ToonyDays's user avatar
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2 answers
54 views

How to simplify complex logs

I have f(x) = (x+$\sqrt{x}$)$\log_2x$ and g(x) = x$log_2(x+\sqrt{x})$. How would I go about simplifying them and obtaining the limit at infinity of f(x)/g(x). So far, the best I have gotten for f(x) ...
elguero's user avatar
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4 votes
1 answer
93 views

Find value of this sum

Let $$\lim_{x\rightarrow 0}\frac{f^{}(x)}{x}=1$$ and for every $x,y \in \mathbb{R} $ we have: $$f(x+y)=f(x)-f(y)+ xy(x+y)$$ Now Find : $$\sum_{i=11}^{17}f^{\prime} (i)$$ I think this question is ...
amir bahadory's user avatar
1 vote
0 answers
44 views

Limit of a Function Involving Hurwitz Zeta Function

I am trying to prove the following limit of a function involving the Hurwitz Zeta function: $$ \lim_{N \to \infty} \frac{\zeta(-d, 1 + N) - \zeta(-d, 1 + p N)}{N^{1 + d}} = \frac{-1 + p^{1 + d}}{1 + ...
Amirhossein Rezaei's user avatar
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0 answers
37 views

If $f'$ is defined on a closed interval $[a,b]$, can we conclude that $f'$ is bounded?

A read a comment by a reputable user that gave me pause. The claim implied that if $f'$ is defined on a closed interval $[a,b]$, we cannot conclude that $f'$ is bounded. Taking this argument in the ...
S.C.'s user avatar
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1 answer
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Double limits. What am I not getting here?

So we learned the epsilon-delta definition of a limit in calculus class. If a limit doesn't exist, it can't be proven this way. I tried this on a limit that we know doesn't exist. I think I've ...
Shakthi Weerawansa's user avatar
2 votes
1 answer
38 views

How can i show this infimum equality $A_{s-}^{-1}=\inf\{t\geq 0: A_t\geq s\}$?

Let $s\mapsto A_s$ be an increasing right continuous function, $s\geq 0$. Then define $A_s^{-1}:=\inf\{t\geq 0: A_t>s\}$ with the convention $\inf\{\emptyset\}=\infty$. By definition the map $s\...
user123234's user avatar
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-3 votes
2 answers
92 views

Proving $\lim \frac{\sqrt[n]{n!}}{n} = \frac{1}{e}.$ [duplicate]

Proving $$\lim_{n\rightarrow \infty} \frac{\sqrt[n]{n!}}{n} = \frac{1}{e}.$$ This is Exercise 12.14 in Ross's Elementary Analysis. It is in the limsup and liminf, and convergence tests chapter. The ...
Arpan Dasgupta's user avatar
2 votes
1 answer
57 views

Confusion about atan vs atan2

I have the following function: $$f(\omega) = \arctan\left(\frac{-\omega\cdot R / L}{-w^2 + 1/(C\cdot L)}\right)$$ When I try to plot it (atan(f(w))), I get the ...
Martel's user avatar
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1 answer
35 views

find limit without using Cauchy's Definitions

I'm trying to find the limit of this: $\lim_{x\to \infty}x^\frac{2}{3}$$[\sqrt[3]{x+1} - \sqrt[3]{x}]$ since I'm not allowed to use any of Cauchy's definitions of the limit, i tried to divide by x. =$\...
S. M's user avatar
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2 votes
3 answers
118 views

Evaluate $\lim_{x\to0^+}\int_{ax}^{bx}\frac{\tanh y\cdot\tan^2y}{y^4}\,\mathrm{d}y$

I've came across an interesting exercise about calculus. For all $x \neq 0$, let $$ f(x) = \int_{ax}^{bx} \frac{\tanh y \cdot \tan^2 y}{y^4} \ \mathrm{d}y,$$ where $a$ and $b$ are positive real ...
Leonardo Javier Stupnicki's user avatar
2 votes
2 answers
36 views

Exchanging limit and inferior limit

Let $b(k,M)$ be a real sequence such that for any $k$, $b(k,M)\in [-M,M]$. I know that $\lim_{M\to+\infty} \sup_{k\geq0} |b(k,M)-a(k)|=0 $, meaning that $b(k,M)$ converges to $a(k)$ when $M\to+\infty$...
Jaz's user avatar
  • 23
2 votes
2 answers
91 views

How do you find the multivariable limit $\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt x +\sqrt y }$

$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt x+\sqrt y}$ considering domain $\{(x,y) \in \mathbb{R}^2 : x,y \ge 0, (x,y) \ne 0\}$ I tried using polar coordinates, but the theta function is unbounded. I also ...
funny0619's user avatar
3 votes
2 answers
70 views

Why does Im($(-1/10^n)^{-1/10^n}$) turn into the digits of pi as integer n gets larger?

$(-.001)^{-.001} \approx 1.007 - .0031634i$, $(-.000001)^{-.000001} \approx 1.000014 - .00000314164$, and $(-.000000001)^{-.000000001} \approx 1.0000000207 - .00000000314159271i$. Notice that as we ...
Alexandra's user avatar
  • 355
0 votes
0 answers
27 views

Limit computation issue

Consider the following function: \begin{equation} Y_t = A_t \left[\delta K^{\frac{\sigma -1}{\sigma}}_t + (1-\delta)L^{\frac{\sigma -1}{\sigma}}_t \right]^{\frac{\sigma}{\sigma -1}} \end{equation} ...
Maximilian's user avatar
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0 answers
41 views

To what points can $f(x,y)=\frac{x^{5}}{\sin^{2}x + \cos^{2}y}$ be extended by continuity?

Considering the function $f(x,y)=\frac{x^{5}}{\sin^{2}x + \cos^{2}y }$, I found that its domain $D_f$ is: $D_f = \mathbb{R}^{2}\backslash \{k\cdot \pi \;|\; k \in \mathbb{Z} \} \times\{k\cdot \frac{\...
Alex's user avatar
  • 55
-2 votes
1 answer
66 views

Evaluate $\lim_{n\to\infty}(\frac{n^2+5}{1+3n^2})^n$.

I came across a question I couldn't solve. The question is: Evaluate $\lim_{n\to\infty}\left(\frac{n^2+5}{1+3n^2}\right)^n$. (This question is after the subsequence section so they want me to find a ...
Someguyalive's user avatar
-3 votes
2 answers
75 views

L'hopital's rule doesn't work

To compare growth rates of $\sqrt{\log{(n)}}$ and $n^a$,$a = 10^{-4}$, I computed the limit $\lim_{n\to\infty} \frac{\sqrt{\log{(n)}}}{n^a}$. Using L'Hôpital's rule, this limit came out to be $0$. ...
Lavender 's user avatar
0 votes
2 answers
105 views

Why are $\sin x$ and $\cos x$ continuous on $\mathbb R$

I have seen many calculus books to use the limits $\lim_{x\to0}\sin x=0$ and $\lim_{x\to0}\cos x =1$ without their proofs. Are these limits obvious? I think no. These two limits are essentially ...
prashant sharma's user avatar
1 vote
1 answer
127 views

Why does $f(t)\le g(t)$ imply $\int_a^b f(t)\; dt\le \int_a^b g(t)\; dt$?

I've come across a proof of the classic limit definition of $e = \lim_\limits{n \to \infty}(1 + \frac{1}{n})^n$ that starts with letting $t$ be any (real, I'm assuming) number in the interval $[1, 1 + ...
Riccardo Iorio's user avatar
0 votes
2 answers
84 views

Does $\lim_{x \searrow 0} \frac{\int_{x^4}^{2x^4} f(y) \ dy}{x^4}$ converge?

I have problems with this exercise: Let $f: [a, b] \to \mathbb{R}$ be continuous. Does $$ \lim_{x \searrow 0} \frac{\int_{x^4}^{2x^4} f(y) \ dy}{x^4} $$ exist? If so determine the limits value. I ...
Felix Gervasi's user avatar
1 vote
0 answers
41 views

Finding $ \lim_{n \to \infty } \frac{1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!}}{e^n} $ [duplicate]

Find this limit: $$ \lim_{n \to \infty } \frac{1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!}}{e^n} $$ My friend managed to solve this through some probability theory (via Poisson limit theorem and ...
  mozhayka's user avatar
2 votes
2 answers
176 views

Compute the limit of the Log-Sum-Exp function

I am trying to prove that the Log-Sum-Exp function converges to the maximum function, i.e. $$ lim_{\tau\rightarrow0}\tau\log\left(\frac{1}{N}\sum_{i=1}^N\exp\left(\frac{x_{i}}{\tau}\right)\right) = \...
rcescon's user avatar
  • 23
-1 votes
0 answers
48 views

Which one of the following is the value of $\lim_{\epsilon\rightarrow0} \int_0^{\infty}e^{\frac{-x}{\epsilon}}(cos(3x)+x^2+\sqrt{x+4})dx$?

Which one of the following is the value of $\lim_{\epsilon\rightarrow0} \int_0^{\infty}e^{\frac{-x}{\epsilon}}(cos(3x)+x^2+\sqrt{x+4})dx$? (1) The limit dose not exist. (2) The limit exists and is ...
Anwar's user avatar
  • 117
1 vote
1 answer
43 views

Limit of $∞.0$ form of an integral and Riemann sum

I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$ where k is chosen such that the order ...
Cognoscenti's user avatar
0 votes
2 answers
106 views

$\lim_{x\to 1} \sqrt{x^2 + 8} = 3$ -- Proof Via Epsilon-Delta Definition of a Limit

This problem was originally asked here: $\lim_{x\to 1} \sqrt{x^2 + 8} = 3$ prove this using epsilon delta. It got closed and is no longer accepting answers. Prove this using epsilon-delta $$\lim_{x\...
Lucien Jaccon's user avatar
2 votes
5 answers
114 views

Let $a_n$ be the sequence defined inductively by $a_1 = 2$ and $a_{n+1} =(1/2)(a_n+2/a_n)$. Prove that $(a_n)^2 ⩾ 2$ for $n ∈ N$.

I have already shown that $a_n ∈ [1, 2]$ for all $n ∈ N$, and I've also been told to start the proof by assuming $a_n < 2$ and finding a contradiction but I have no clue how to do this.
mathmath's user avatar
2 votes
0 answers
76 views

If $\lim\limits_{h\to0 } f(x+h)- f(x-h)= 0$ how to prove that there are at most countable set of discontinuities [duplicate]

I saw this problem : Suppose $f$ is a real function defined on $\mathbb{R}^1$ which satisfies $\lim\limits_{h\to0 } f(x+h)- f(x-h)= 0$ for every $x \in \mathbb{R}^1$ Does this imply that $f$ is ...
pie's user avatar
  • 3,457
1 vote
1 answer
54 views

Limit of difference of two functions

I've come across different proofs for one of the limit laws, namely, $$lim_{x\to c} [f - g](x) = lim_{x \rightarrow c}\ f(x) - lim_{x \rightarrow c}\ g(x) .$$ Now, some authors end their proof like so,...
FeedMePi's user avatar
  • 107
-3 votes
0 answers
26 views

Given that that the integrals of f^4(x) and that of |f(x)| converge, prove that the integral of f^2(x) converges. (Integrals are from 1 to infinity) [closed]

I need to prove that if the integral from 1 to infinity of f^4(x) and the integral from 1 to infinity of |f(x)| converges, then the integral of f^2(x) from 1 to infinity converges. f is of course ...
Ori Dahan's user avatar
1 vote
1 answer
74 views

Limit involving an Indicator Function and Sum

For each $n\in\mathbb{N}$, consider real $n$ numbers $x_{i,n}$, $1\le i \le n$, given by $$\sum_{i=1}^{n}x_{i,n}=0,\\-1/n \le x_{i,n} \le 1-1/n.$$ I am trying to find value of $$\lim_{n\to \infty} \...
The Substitute's user avatar
0 votes
0 answers
48 views

Is $\lim_{t\to \infty} \frac{f(t+o(t))}{f(t)}=1$ for bounded continuous function $f$?

Let $o(t)$ denote any continuous function with $\lim_{t\to \infty} \frac{o(t)}{t}=0$. I wonder if $\lim_{t\to \infty} \frac{f(t+o(t))}{f(t)}=1$ for bounded continuous function $f$ on $\mathbb R$. The ...
taylor's user avatar
  • 557
0 votes
2 answers
106 views

Standard limits giving wrong answers.

As we all know, $$ \lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)=1 $$ And I encountered many questions where I used this standard limit and it gave the right answer. However in the following ...
Sanket 's user avatar
-2 votes
0 answers
35 views

Standard limit causing wrong answer of limits. [closed]

By using standard limits, many of the questions give zero as answer but the answer is not zero, when we use expansion method. Please help. In this question, if I directly use standard limit(sinx/x =1),...
Sanket 's user avatar
1 vote
1 answer
42 views

How important is the order of the conditional in the ($\epsilon$, $\delta$)-definition of the limit? Would it matter if it is a biconditional?

So we're looking at the ($\epsilon$, $\delta$)-definition of the limit in class and I am kind of confused, because the teacher says one thing but the books say another. Teacher says that $\lim_{x \to ...
zlaaemi's user avatar
  • 1,009
0 votes
1 answer
27 views

Lebesgue integral and limit; $\lim_{a\to 0^+}\int_{a}^{1}(t\ln(t))^3dt=\int_{0}^{1}(t\ln(t))^3dt$

How can I prove: $$\lim_{a\to 0^+}\int_{a}^{1}(t\ln(t))^3dt=\int_{0}^{1}(t\ln(t))^3dt$$ assuming that the integral are Lebesgue integrals. There may be a theorem that confirms this equality.
Minimus Heximus's user avatar
0 votes
0 answers
73 views

Convergence of $\lim_{K\rightarrow +\infty}(\sum_{k=1}^{K} a_k^2) / (\sum_{k=1}^{K} a_k)$?

Whether $(\sum_{k=1}^{K} a_k^2) / (\sum_{k=1}^{K} a_k)$ can converge to $0$ as $K\rightarrow \infty$ if $a_k > 0$, $a_k\rightarrow 0$ but $\sum_{k=1}^{\infty} a_k \rightarrow +\infty$? If take $a_k ...
Yuzhen Feng's user avatar

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