Questions on the evaluation and properties of limits.

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Help with $\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1})$

$\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1}) = ?$ I don't know how to solve the indetermination there... is it possible to rearrange the expression in brackets in order to use ...
1
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3answers
74 views

How to find $\lim\limits_{n\to \infty }(1 + \frac{1}{n})^n$

$\lim _{n\to \infty }\left(1 + \frac{1}{n}\right)^n$ How do I get started with this one? Variable substitution would be one way, but our lecturer hasn't covered that yet, so there should be some ...
2
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1answer
42 views

Can this special case happen when working with L'Hopitals rule?

I am using this version of L'Hopital's rule Assume that $\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}g(x)=0$, and that the limit-value $\lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ exists (could ...
2
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4answers
54 views

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$.

Find $\lim_{x\to \infty}{[({1\over e}(1+{1\over x})^x)]^x}$. I have been trying for hours using the continuity of $e$ and using L'Hopital rule but it gets really scattered and ugly. I am in despaire. ...
1
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2answers
127 views

Limit under integral sign

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
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1answer
30 views

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. What has gone wrong?

Find $\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}$. Using L'Hopital's rule I get that $$\lim_\limits{x\to 1}{x^\alpha-1\over x^\beta -1}=\lim_\limits{x\to 1}{\alpha x^{\alpha-1}\over \beta ...
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1answer
32 views

Interpreting limit notations

My question is: Are the following notations equivalent or not: $$(1)\;\;\;\;\;\;\text{When}\;||\textbf{x}||\rightarrow 0,\;\text{then}\;\;\;\frac{f(\textbf{x})}{||\textbf{x}||}\rightarrow0$$ ...
4
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2answers
58 views

Limit $I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$

Im a new participant in this mathematical forum, so this is one of that i couldn't solve it. $$I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$$ I've ...
4
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1answer
52 views

Solve $(\alpha,\beta)$ for $\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$

Find the ordered pair $(\alpha,\beta)$ with non-infinite $\beta \ne 0$ such that $$\lim_{n\to\infty} \frac{\sqrt[n^2]{1!2!\cdots n!}}{n^\alpha} = \beta$$ My approach: $$\ln (1!2!\cdots n!) = (n)\ln ...
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0answers
29 views

Sanity check on limit problem

I'm second guessing myself on limit problem. Is this all ok using continuity of log & exp + calculus of limits inside of log to get to 2/3 ? ...
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0answers
26 views

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. [duplicate]

Find $\lim_\limits{x\to \infty}{\sqrt[3]{x^3+ax^2+bx+c}-x}$. As I understand, I should use Taylor series, but I don't know how. What should I translate into Taylor series, to what extent, etc. This is ...
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0answers
21 views

Finding the limit of the series

Find the limit of the sequence , can somebody help ,I'm struck.
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0answers
29 views

peicewise fuctions with limits [on hold]

$$f(x)= \begin{cases} 0 & \text{if } x \text{ is rational}\\1 & \text{if } x \text{ is irrational}\end{cases}$$ and $$g(x)= \begin{cases} 0 & \text{if } x \text{ is rational}\\x & ...
6
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1answer
58 views

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$

Find $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$. I do know it is bounded by $1$. I tried using the sandwich rule with no ...
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1answer
17 views

Convergence of sequence depending on parameter

How can I be more rigorous in solving the following than just by inspection. $$\lim_{n\to\infty}\frac{(n+1)^\alpha}{(3n+3)(3n+2)(3n+1)}$$ It's plain to see that if $\alpha>3$ it diverges and for ...
1
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4answers
64 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
0
votes
1answer
23 views

Dyadic expansion

I'm reading the appendix in Billingsley book "Probability and measures" and I can't understand the following. If $$\sum_{i=1}^n\frac{d_i(\omega)}{2^i} < \omega \leq ...
2
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5answers
79 views

Limit problems and quandaries: finding $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$.

Find $\lim_\limits{n\to \infty } {({n^2-n\over n^2+1})^{n+10} }$. What I did is: $\lim_\limits{n\to \infty }{({n^2-n\over n^2+1})^{n+10}}=\lim_\limits{n\to \infty } {({n^2+1-1-n\over ...
4
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2answers
193 views

Limit at Infinity: $ \lim\limits_{n\rightarrow\infty}n\left(1-\frac{1}{\ln(n)}\right)^n$

Maple says that this limit is zero but I can't prove it. Any help or tips would be appreciated. $\displaystyle\lim_{n\rightarrow\infty}n\left(1-\frac{1}{\ln(n)}\right)^n$
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2answers
37 views

Limits and derivatives - limit of a trigonometric function

Question: Find the limit of: $$\lim_{x \to 0} \frac{\sin^9x}{x}.$$ Well using the L-Hopital's Rule, I can write: $$\lim_{x \to 0} \frac{\sin^9x}{x} = \lim_{x \to 0} \frac{9\cos^8x}{1}.$$ This ...
2
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0answers
70 views

Question about proof particular L'Hospital's case

My brain is not exactly understanding a particular proof for the L'Hospital's case when $x$ goes to infinity. The author considers $\lim\limits_{x\to+\infty}\frac{f(x)}{g(x)}$ where he subs $t=1/x$ It ...
7
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1answer
115 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: i tried to evaluate the integral $$\begin{align} ...
1
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1answer
47 views

${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$

Let $f\in \mathbb{R}^{I}$ $I$ interval of $\mathbb{R}$ Show that $${f \text{ is differentiable on } I \iff f_{\left|\ [a,b]\right.} \text{ is differentiable }\ \forall a,b \in I}$$ in ...
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0answers
20 views

Evaluate the limit of series [on hold]

Finding the limit of series Sigma [r^1/8(n^x-1/x +r^x-1/n)]/n^x+1 as n tends to infinity , where r can take values of natural numbers and x is a constant.
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1answer
30 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
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0answers
18 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
1
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2answers
26 views

Choosing path to show limit does not exist

I'm trying to show that the limit as $(x,y)$ go to $(0,0)$ for the function $f(x,y) = sin( x + y )/( |x| + |y|)$ does not exist. I initially tried the path $y=2$ and $y=1$, but I don't think I can use ...
3
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1answer
53 views

Find $\lim_\limits{n\to \infty}\{en!\}$.

Find the limit $\lim_\limits{n\to \infty}\{en!\}$. $Attempt:$ $\lim_\limits{n\to \infty}\{en!\}=\lim_\limits{n\to \infty}\{(1+{1\over 1!}+{1\over 2!}+{1\over 3!}+...+{1\over n!}+...)n!\}$. The ...
5
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3answers
39 views

Finding $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$

Find $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$. I tried using l'Hospital rule with the continuity of $e$ function. Also tried using Taylor expansion with no success. What ...
0
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1answer
86 views

Theorem $4.3.12$ on ( Mathématiques en BCPST Tome 1 Pascal BEAUGENDRE )

Let $f\in \mathbb{R}^{I}$ and $x_0 \in \stackrel{\ \circ}{I} = \mathrm{int}(A)$ Show that $$\left.\begin{matrix} f \text{ is continuous at } x_0 \\ f \text{ is differentiable at all } x \in I ...
3
votes
2answers
148 views

Limit of solution of differential equation without solving the equation.

Given $$x'(t)=A-B\left(x(t)\right)^2, \quad x(0)=0.$$ Is it possible to find $\lim\limits_{t\to\infty}x(t)$ without solving the differential equation? Assuming $\lim\limits_{t\to\infty}x'(t)=0$ gives ...
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3answers
30 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
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3answers
55 views

Explanation for $\lim_{x\to2} e^{\frac{1}{x-2}}$

I can't find out why is the limit from the left side = 0 and from the right = Infinity?
3
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1answer
71 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
5
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1answer
58 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
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0answers
21 views

Proving the following multivariable limit using the definition. [on hold]

I am trying to prove that $ \lim_{(x, y) \to (0, 0)} x + 2xy + 2y + 6 = 6$ using the definition of a multivariable limit, but I am having no luck. Could someone please help me? Thanks!
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2answers
33 views

Negate definition of limit [on hold]

The definition for $\lim \limits_{x\to a} f(x) = L$ is the following: For all real numbers $\varepsilon > 0$, there is a real number $\delta > 0$ such that for all real numbers $x$ if $a−\delta ...
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0answers
50 views

$\epsilon$-$\delta$ proof of a sinc limit in Complex variables [on hold]

I am stuck on the following problem : Prove (using $\epsilon$-$\delta$) that $$\lim_{z \rightarrow \pi/2} \frac{\sin z}{z} = 2/\pi$$ Basically I do not know how to get an estimate on ...
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4answers
90 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
3
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2answers
30 views

Limit Definition for Proof

How can I use the definition of a limit to set up a proof for a statement such as: $$\lim_{n\to\infty} {f(n)} \to \infty$$ I have tried applying the standard definition, but I come out with ...
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2answers
40 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
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5answers
55 views

Evaluate the Limit $\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$

Evaluate the Limit: $$\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$$ I am trying to understand how to do this. I have to use series expansion and not L'Hospital. Any help would be great. ...
4
votes
2answers
42 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
1
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1answer
37 views

Help with troublesome limit [on hold]

I need help in computing the following limit: $$\lim_{x\to 0} \frac{a}{x}~\exp \left(-\frac{a^2(\log(bx))^2}{2}\right)$$
2
votes
3answers
46 views

$\lim_{x \to 0} \sin x ^{\sin x} $ to determine.

$\lim_{x \to 0} \sin x ^{\sin x} $ Hi, Help me do it please. I am asking for any advices, helpful observations. Thanks in advance.
0
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1answer
33 views

Functions and continuity proof in real analysis

Prove: If $f\colon A\rightarrow\mathbb{R}^m$ and $a\in A$, show that $\lim_{x\rightarrow a}f(x)=b$ if and only if $\lim_{x\rightarrow a}f^i(x)=b^i$ for $i=1,\dots,m$. The end of the statement is ...
2
votes
1answer
31 views

How to calculate the limit of this sequence which incorporates tan?

I was revising for my pre-calculus exam, which is in two weeks time, and I started proving some sequence related theorems. I got interested in limits and I started deepening the concept. I got to a ...
1
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3answers
39 views

How to convert into a definite integral

Could you show me how to convert the following into a definite integral: $$\lim\limits_{n \to \infty} \sum_{k=1}^{3n} \frac{1}{n}\cos\left(\frac{k\pi}{n}\right)\sin\left(\frac{2k\pi}{n}\right)$$ ...
0
votes
2answers
53 views

sum of exponentials to non-integer power

I have the expression \begin{equation} (e^{at}+e^{bt}+e^{ct})^{v} \end{equation} for some a,b,c which isn't important. I'd like to take a limit $t\rightarrow \infty,v\rightarrow 0,vt=\text{constant}$. ...
2
votes
3answers
65 views

Limit evaluation with integral

Evaluate the limit $$\lim_{n\to\infty} \int_0^1 n^2x(1-x^2)^n dx$$ My Proof: We may look at $n$ as a constant and evaluate the integral $\int_0^1 x(1-x^2)^ndx$ (I already moved out the $n^2$). ...