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 (limits-colimits) instead.

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For each set of conditions below give a formula for a function that satisfies the conditions or explain why one cannot exist.

(a) A function with exactly two horizontal asymptotes and exactly two vertical asymptotes, but is defined everywhere else on $\mathbb{R}$. (b) A continuous function on $\mathbb{R}$ with $f(2)=−3$, $f(−...
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Calculate the following limit: $\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$ [closed]

Hi I've been stuck trying to calculate this limit, I'd appreciate the help. $$\lim\limits_{x\to\infty}\frac{\left(1+\cos\left(2^{-x}\right)\right)^x}{2^x}$$
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3answers
22 views

Confirmation of polar coordinate calculation

$$\lim_{x,y \to 0,0} \frac{x^2y + xy^2}{ x^2+y^2}$$ Polar coordinates $$x= r \cdot \cos \theta$$ $$y = r \cdot \sin \theta$$ $$\lim_{r \to 0} \frac{(r \cos \theta)(r \sin \theta) + (r \cos \theta) (r \...
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2answers
79 views

Help with $ \lim \frac{n\pi}{4} - \left( \frac{n^2}{n^2+1^2} +\frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) $

I have proved that $$\lim_{n\to \infty} \left( \frac{n}{n^2+1^2} +\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4},$$ by using the Riemann's sum of $\arctan$ on $[0,1]$. Now I'...
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0answers
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How can I calculate the time for a payload (quadruple drone ) falling starting from 0m/sec then reaching to 9m/s descending with 9m/s from height 250m

I have the velocity in descending order but I want to have the velocity time graph if anyone could help me I will be thankful.
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About set of subsequential limits

Let $\left\{x_{n}\right\}$ be a sequence in the $((0, 1) \cap \mathbb{Q} ) \subset \mathbb{R}$ with following properties: 1.$\left|x_{\text {n+1}}-x_{n}\right|<\frac{1}{n} ,n \in \mathbb{N}$ 2.$\...
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1answer
17 views

Limit supremum and differentiability of power series

There is this part in Newman and Bak's Complex Analysis where he justifies via lim sup, that the series obtained by differentiating a convergent series has the same radius of convergence as the former ...
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1answer
31 views

Monotone convergence theorem for a generic $f(x,y)$ instead of $f_n(x)$

I was wondering if it is safe to say that the monotone convergence theorem $$\lim_{n\to\infty} \int_X f_n(x) = \int_X \lim_{n\to\infty} f_n(x)$$ (where $f_n(x)$ is a non-decreasing sequence, etc.) is ...
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Problem with a limit (no L'Hospital) [closed]

I'm stuck with this limit and I don't know how to solve it. $$\lim_{x \to 3} \frac{(x^2-9)^{1/2}-(x-3)}{e^{-1/(x^2-9)}}$$ I'm not allowed to solve it the via Taylor-McLaurin expansion or L'Hospital. ...
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Probable existence of an almost integer contained in a limit

I found this almost integer in studying the limit : $$\lim_{x\to \infty}\Gamma\left(\sin^2\left(\frac{1}{x}\right)\right)\Gamma\left(\sin\left(\frac{1}{x}\right)\right)-x^3=-\infty$$ Well my goal was ...
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Using the definition of limit to show $\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$ [closed]

I need prove this limit by the formal definition of limit: $$\lim_{x\to -3}\frac{x-3}{x^2 -9} = \frac{-1}{6}$$
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【If $f $ is NOT a polynomial ( or a fraction of polynomials), then we might have $\lim_{x\to a} f(x) \ne f(a)$.】How to understand this sentence?

【If $f $ is NOT a polynomial ( or a fraction of polynomials), then we might have $\lim_{x\to a} f(x) \ne f(a)$.】How to understand this sentence? The following is my understanding,Are they correct or ...
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3answers
68 views

What is the limit of $\frac{e^n}{(n+4)!}$

How do we compute the limit: $\lim_{n\rightarrow \infty} \frac{e^n}{(n+4)!}$ I can only think of this approach, but I am not sure that it's too valid: We know that the terms are positive, so we would ...
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1answer
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Solve limit without L'Hopital (unknown function)

I'm trying to do an exercise but I honestly don't know what to do to solve it. I know this will be a very basic question, but I'm learning limits. The exercise is: Given: $$\lim\limits_{x \to 2} \frac{...
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1answer
28 views

Prove: $\lim_{x\to a}\lvert f(x) \rvert=L \rightarrow\Big[\lim_{x\to a} f(x) =L \lor \lim_{x\to a} f(x)=-L \Big]$, where $L \geq 0$

I am trying to prove the following statement: $\displaystyle \lim_{x\to a}\lvert f(x) \rvert=L \rightarrow\Big[\displaystyle \lim_{x\to a} f(x) =L \lor \displaystyle \lim_{x\to a} f(x)=-L \Big]$, ...
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3answers
48 views

Why do we call a limit that evaluates to $\displaystyle\pm\infty$ a particular instance of “The Limit Does Not Exist”

The $\epsilon$-$\delta$ definition of a limit is stated as: $\displaystyle \lim_{x \to a}f(x)=L \iff \forall \epsilon \gt 0\ \exists\delta \gt 0 \text{ s.t. } \forall x \in \mathbb R \big [ 0\lt \...
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1answer
62 views

Conjecture regarding sequence of improper integrals of form $\int_1^\infty \frac{\ln x}{f(x)}dx$

I was playing around with integrals of the form $$\int_1^{\infty}\frac{\ln x}{f(x)}dx$$ and noticed something interesting when $f(x)=x(x-1)$. By numerical computation the answer tends to $1.644\ldots$ ...
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0answers
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Proof of L'Hopital's rule for complex functions

I was given the task to prove that if: $$\lim_{z\to z_0}f(z)=\lim_{z\to z_0} g(z)=\infty$$ and: $$\lim_{z\to z_0}\frac{f(z)}{g(z)}$$ exists, then the following is true: $$\lim_{z\to z_0} \frac{f(z)}{g(...
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3answers
51 views

evaluating limit of $\lim_{x \to 0}\frac{1-\cos(x)}{\sin(x)(e^x-1)}$

This is for evaluating limit of $\lim_{x \to 0}\frac{1-\cos(x)}{\sin(x)(e^x-1)}$. It's easy with evaluating using L'Hôpital's Rule, but I want to use Taylor series. I can see here $\cos(x)=\sum_{n=0}^{...
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1answer
110 views

Proving $\lim\limits_{x\to0}{\frac{\sin(\frac{1}{x})}{\sin(\frac{1}{x})}}=1$

I am relatively new to calculus and I'm trying to understand it rigorously. For this question, assume that I only consider the functions to have purely real domains and ranges. I have seen that $\lim\...
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3answers
36 views

Evaluate $\lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})}$ without l'Hospital

$$ \lim_{h \to 0} \frac{\sin(\frac{h}{2})-\frac{h}{2}}{h\sin(\frac{h}{2})} $$ I've worked the last few hours on this equation and still didnt find a way to evaluate it. The idea I had was to bound ...
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2answers
46 views

If for every positive number $\varepsilon$ there exists a positive number $\delta$, why not the other way back? [duplicate]

Definition. Let $f(x)$ be a function of $x$. If for every positive number $\varepsilon$, however small it may be, there exists a number $\delta$ such that whenever 0 $<$ |x - a| $<$ $\delta$ we ...
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0answers
17 views

Taking a limit under absolute value

Assume $Z = \mu V + \sigma \sqrt{V}U$, where $V \sim \Gamma(n/2,1/2)$ and $U$ is a sum of dependent standard normal random variables. I'm looking at $\frac{|Z|}{n}$, as $n \to \infty$. My question: is ...
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2answers
53 views

Find the limit of following sequence

$\lim\limits_{n \to inf}\frac{n×1^r+(n-1)×2^r+...+1×n^r}{n^{r+2}}$ Ok so the only idea I got was to use Stholz theorem and then use $1^r+2^r+...+n^r$ ~ $ \frac{1}{(r+1)} (n^{r+1})$ somehow but that ...
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2answers
32 views

What's the difference between approaching $0$ and becoming $0$ in this context?

My calculus book, Differential and Integral Calculus by Piskunov, states the following theorem: This theorem is used in evaluating the following limit: $$\lim_{x \to 1}\dfrac{x}{1-x}$$ We first take ...
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0answers
41 views

Why Wolfram Alpha took the constants apart and then evaluate the limit? Can't i just merge Ln(1+e^3) with Ln(1+e^x) using division rule?

**So i tried my method where i merged both log (Ln) and i got different answer why? ** That's my method btw :
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2answers
41 views

Evaluating the following limit without L'Hopital's help

I am given the following limit $$\underset{x\to 0}{\mathop{\lim }}\,{{\left( {{cosx}} \right)}^{{-x}^{-2}}}$$ I tried using the fact that $cosx=1-2sin^2(\frac{x}{2})$, but it didn't give me anything. ...
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0answers
19 views

Limits and existence of decay rates

My question is perhaps elementary, still I'm neither managing to figure out counterexamples nor to prove the result. Let $(x_n)_{n\geq 1}$ be a real-valued sequence and assume that $x_n\to x\in \...
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0answers
33 views

Limit of an integral where (in my opinion) it is not possible to apply de l'Hopital rule

I want to prove that the following limit is $0$: $$\lim_{x\to 0}\frac{1}{x}\int_0^{x^3}f(t)\,dt$$ The only hypothesis of the problem are: $f$ is bounded and integrable. So I have thought that since I ...
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2answers
60 views

Alternating series: Does $\sum\limits_{n=2}^{\infty} (-1)^n \frac{1}{e^{\ln(n)}}$ converge?

To be specific I need to determine whether the series absolutely converges or conditionally converges. I have already determined if the series converges and diverges (it converges) through the ...
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0answers
19 views

Finite expansion of sin$(\delta)$cos$(\beta-\delta)$ [closed]

In the finite expansion of sin$(\delta)$cos$(\beta-\delta)$ ($\beta<<1, \delta<<1$), what should we do with the big Os, i.e. $O(\delta^3)O((\beta-\delta)^2) $, and $O(\delta^3)+O((\beta-\...
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Prove that the $\epsilon$-$\delta$ definition of a limit is equivalent to this other definition

I'm reading Differential and Integral Calculus by Piskunov, and found this another definition of limit (along with the $\epsilon$-$\delta$ definiton): The topic just ends by stating that it is easy ...
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1answer
49 views

$1 - \lim_{x \to \infty} \left( 1 - \frac{2}{x^2} \left( 1 - \exp \left( -\frac{x^2}{2} \right) \right) p \right)^{kx^2}$

I'm currently working on my master thesis and I need to solve this limit. I forgot almost everything about limits since last time a I saw them was basically in high school. Anyway, I solved this limit ...
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2answers
30 views

How to evaluate limits of these functions

Good day, I am currently trying to calculate the show the following. Given the function $f(x)=(\alpha_1 x_1 ^{\rho} + \alpha_2 x_2 ^ {\rho})^{1/\rho}$, where $\alpha_1+\alpha_2=1$. I am trying to ...
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2answers
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Manipulating limit expressions when you are told that the limit of one of the expressions does not exit (Spivak - Chapter 5 Problem 23)

Questions 23 a) and b) from Chapter 5 of of Spivak's Calculus are written as follows: a) Suppose that $\displaystyle\lim_{x \to 0}f(x)$ exists and is $\neq 0$. Prove that if $\displaystyle\lim_{x \to ...
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1answer
37 views

Limit with integral, case infinity * 0 [duplicate]

I attached an image with a limit, how can I solve that limit? If I use mean value theorem for integral it will be indeterminate form infinity * zero.
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0answers
23 views

Limit of matrix power tend to eigenvector

I wish to show the following for a non-negative matrix $K$ by this i just mean each entry is bigger than or equal to $0$. $$\lim_{k\rightarrow\infty}\frac{K^kv}{\|K^kv\|}=\pm e_1.$$ The $v$ here is ...
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1answer
66 views

Showing $\lim_{p\to\infty}\left(\int_I|f|^p\right)^{1/p} = \max|f|$, where $I$ is a generalized rectangle

Let $I$ be a generalized rectangle and let $f: I \to \mathbb{R}$ be continuous. Show that $$\lim_{p\to\infty}\left(\int_I|f|^p\right)^{1/p} = \max|f|$$ I found it straightforward to show that these ...
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4answers
95 views

How to find $ \lim_{x \to \frac{\pi}{2}}\frac{\cot^{2}x }{1-\sin x} $?

I tried to use cosec identities to solve, but I am having trouble especially with the cot identity. The closest I got to the answer is $$ \frac{1-\sin^{2}x}{(\sin^{2}x - \sin^{3}x)} $$ which I don't ...
2
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1answer
48 views

limits calculation /product

Calculate $\;\lim\limits_{n \rightarrow \infty}\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\ldots\left(1-\dfrac{1}{n}\right)$ What I have done: $\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{...
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2answers
52 views

Calculate limits without l'Hôpital

I am trying to calculate the following limits $$ \lim_{x\to\infty}(2x+1) \ln \left(\frac{x-3}{x+2}\right) $$ and $$ \lim_{x\to1}\frac{x}{3x-3}\ln(7-6x) $$ I can't use l'Hôpital's rule, so I am not ...
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4answers
90 views

Evaluate $\underset{x\to 0+}{\mathop{\lim }}\,{{\left( {{x}^{x}} \right)}^{x}}$

We’ve already proven $\underset{x\to 0+}{\mathop{\lim }}\,{{x}^{x}}=1$ in the classroom. Here’s my quick and dirty attempt: $\underset{x\to 0+}{\mathop{\lim }}\,{{\left( {{x}^{x}} \right)}^{x}}={{\...
2
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4answers
55 views

Limit calculation with root

Let $x, y \in \mathbb{R}$. Prove that $\lim_{n\rightarrow \infty} n(1-\sqrt{(1-\frac{x}{n})(1-\frac{y}{n})} )=\frac{x+y}{2}$ I tried some stuff but somehow I can not manage to find the right direction ...
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3answers
51 views

$\epsilon$-$\delta$ definition about the limit, is possible to change $\forall \epsilon>0$ to $\exists\epsilon>0$

click for link What would happen if $\forall\epsilon>0$ in the definition change into $\exists\epsilon>0$. Why is it not correct?
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3answers
62 views

What is wrong with this proof of $e \le 2$

For every $n$, $e \le \left(1 + \frac{1}{n}\right)^n$. $$ \left(1 + \frac{1}{n}\right)^n = 1 + \frac{n}{1!}\times\frac{1}{n} + \frac{n(n-1)}{2!}\times\frac{1}{n^2}+\ldots+ \frac{n!}{n!}\times\frac{1}{...
2
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0answers
36 views

How to deal with big O in finite expansion?

suppose we have $f(x) = \sin x + 2$ and $g(x) = \cos x - 3$. I know that $\sin x = x + O(x^3) $ and $\cos x = 1+ O(x^2)$. Finite expansion of $f(x)×g(x)$ will have some terms like that:$ O(x^3)O(x^2) +...
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0answers
21 views

Is the $L^1$ limit of bounded functions again bounded?

For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$. Assume $f_n\to f$ in $L^1(\mathbb{R})$. Is it true that $|f|\leq 1$ almost everywhere ? I would do it like this: If $f_n\to u$ in $L^1(\mathbb{...
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2answers
121 views

Help evaluating $\lim_{x \to 0}{\frac{a^{k} - a^{k - x}}{bx}}$ without L'Hospital?

Evaluate $\displaystyle\lim_{x \to 0}{\frac{a^{k} - a^{k - x}}{bx}}$ if $a$, $b$, and $k$ are positive real numbers. For this attempt, let $L$ be the limit, if it exists. Then, \begin{align*}L &\;...
3
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1answer
62 views

Check my idea: liminf and limsup definitions

Let $N\geq 1$ and let $F:\mathbb{R}^N\to \mathbb{R}$ be a continuous, radially symmetric and nonnegative function such that: $$(i): a\in\mathbb{R} \mbox{ exists such that } \liminf_{r\to +\infty}\frac{...
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1answer
27 views

Prove limit of convergence numerical sequence in infinity special condition

Does anyone know how to prove this problem. If $\{a_n\}$ is a convergent numerical sequence, and $b$ is a real number that appears in infinit number in $\{a_n\}$, then limit of $\{a_n\}$ when $n$ goes ...

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