Questions on the evaluation and properties of limits.

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24 views

Reciprocal of a limit that goes to infinity

Lets say we have a limit $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = +\infty$, then is it safe to assume that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n} = 0$?
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1answer
47 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?
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0answers
10 views

$\pi_n(y)\in O(1)$ for every realisation $y$ of $Y_n$ implies $\pi_n(Y_n)\in O_p(1)$

Consider a sequence of random variables $\{Y_n\}_n$ all defined on the same probability space $(\Omega, \mathcal{F}, P)$ such that $Y_n:\Omega \rightarrow \mathbb{R}$ $\forall n$. Consider a sequence ...
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0answers
9 views

Limits of Kummer Confluent Hypergeometric function for fixed z

I have the following: \begin{equation} \sum_{j=0}^{\alpha_{2}}C\, e^{az}\, M(-\alpha_{2}+j,-\alpha_{3}+j,-\lambda z)\Bigg|_{z=-\infty}^{0} \end{equation} $C$ is a constant, $a,\alpha_{2},\alpha_{3}, ...
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1answer
27 views

Finding limit of $\frac{a_n^3+5n}{a_n^2+n}$ for $(a_n)$ bounded.

Suppose that the sequence $(a_n)_{n \in \mathbb{N}}$ is bounded. Prove that the sequence $(c_n)_{n \in \mathbb{N}}$ defined by $$ c_n = \frac{a_n^3+5n}{a_n^2+n} $$ is convergent and find its ...
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1answer
32 views

How do i evaluate this limit as n goes to infinity $(8n- \frac{1}{n})^{\frac{(-1)^n}{n^2}}$ [on hold]

How do i evaluate this limit as $n \to \infty$: $$\lim_{n\to\infty}\left(8n- \frac{1}{n}\right)^{\frac{(-1)^n}{n^2}}$$ Thanks
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2answers
54 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
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0answers
13 views

Existence of a limit - Composition of continuous functions - Questioning

The question of Jim Darson to this link, Don Antonio replied using a similar property in the composite of continuous functions ($\frac{\text{Re}\,z}z$ and the line $\;y=mx\;$) is continuous, but with ...
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2answers
38 views

What does this limit evaluate to? [on hold]

Please help me evaluate the following limit$$\lim_{n\to\infty}\frac{\displaystyle2\left(\frac{3}{5}\right)^{n+1}-3}{\displaystyle 2\left(\frac{3}{5}\right)^n+3}$$
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2answers
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$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad n\in I$

$$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad(n\in I) \\ \text{where $\lfloor\cdot\rfloor$ is the greatest integer function.}$$ This is what I ...
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2answers
24 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
0
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2answers
23 views

Proving that $f^2$ is differentiable given that f is differentiable at $(x_0,y_0)$

So I've tried using the definition: $f$ is differentiable at $(x_0,y_0)$ iff $$ f(x,y)-f(x_0,y_0)=\frac{\partial f}{\partial x}(x_0)\cdot x+\frac{\partial f}{\partial y}(y_0)\cdot y+o(\sqrt ...
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1answer
36 views

Conjecture: $\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$

I conjecture: $$\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$ Is this provable? If this is false, then can I have a function $f$ such that: $$\lim_{x\to\infty} ...
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2answers
27 views

Is it possible to extend $f$ by continuity at $z = 0$? Why or why not?

Let $f(z) = \frac{z}{|x|}$, with $z \not=0$ (a) Construct two sequences ${u_n}$ and ${v_n}$ such that $\lim_{n \to \infty} u_n = 0$ and $\lim_{n \to \infty} v_n = 0$ $\lim_{n \to \infty} f(u_n)$ ...
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0answers
60 views

Find $\lim_{n\to\infty} \frac{1}{n}\left(1+\frac{1}{2}+\frac{1}{3}+\dotsb+\frac{1}{4n-1}+\frac{1}{4n}\right)^5.$

Find $$\lim_{n\to\infty} \frac{1}{n}\left(1+\frac{1}{2}+\frac{1}{3}+\dotsb+\frac{1}{4n-1}+\frac{1}{4n}\right)^5.$$ I don't know how to calculate this limit. Please show me the how to solve this ...
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4answers
56 views

Solving $\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$

I have trouble finding the value of the following limit: $$\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$$ For now I have rewritten the term into: $$ \lim_{n \to \infty} ...
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2answers
49 views

Evaluating $\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$ without L'Hospital

I have been trying to evaluate $$\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$$ Finally, I used the L'Hospital's Theorem and I got the answer $1/2$, but I wonder if there is a way to solve this without this. ...
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1answer
13 views

How do I prove that the floor function has no limit?

I don't really know where to start. I had this approach in mind: take any integer, let's name it $k$, and show that the limit on the left side of $k$ equals to $k-1$, and that the limit on the right ...
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0answers
22 views

How would I find the limit for this equaiton? [duplicate]

How would you find the limit of this function: The n value is always even (this is for a problem and it would not make sense for n to be an odd number) Also, how could I plot this with n as the x ...
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2answers
32 views

Can I prove a function is continuous by looking at the domain?

I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one ...
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2answers
32 views

Finding the limit of a sequence $\lim_{n \to \infty} 2^{2n+3}\left(\sqrt[3]{8^n+3}-\sqrt[3]{8^n-3}\right)$

I have the following limit $$\lim_{n \to \infty} 2^{2n+3}\left(\sqrt[3]{8^n+3}-\sqrt[3]{8^n-3}\right)$$ I don't know with what I should multiply the term. I know that if I have for example $$\lim_{n ...
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0answers
22 views

$\lim\limits_{K\to\infty} [n,K]$.

I've seen the following notation sometimes $$\lim\limits_{K\to\infty} [n,K]$$ And some people claim this is equal to $[n,\infty)$. They say, well $K$ keeps getting larger so for whatever $x\ge n$, we ...
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votes
2answers
42 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
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2answers
19 views

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist. Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} ...
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1answer
36 views

Find $\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$

$$\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$$ I have multiply by $\frac{\sqrt{2x^2+2}+(x+1)}{\sqrt{2x^2+2}+(x+1)}$ and got: $$\lim_{x\to 1} ...
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0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
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2answers
74 views

Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$

Find: $$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{dx}{x^2 \log{(1+ \frac{x}{n})}}$$ The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ ...
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1answer
35 views

$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$ - show that $a_n$ is convergent sequence

Problem: Show that $a_n$ is convergent sequence and find a limit of $a_n$. $$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$ I tried to look at this as normal limit problem so I wrote ...
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3answers
54 views

Find $\lim_{n \rightarrow \infty} \int_0^n (1+ \frac{x}{n})^{n+1} \exp(-2x) \, dx$

Find: $$\lim_{n \rightarrow \infty} \int_0^n \left(1+ \frac{x}{n}\right)^{n+1} \exp(-2x) \, dx$$ The sequence $\left(1+ \frac{x}{n}\right)^{n+1} \exp{(-2x)}$ converges pointwise to $\exp{(-x)}$. So ...
4
votes
4answers
49 views

limit of form “$∞ \cdot 0$”

I am trying to formally prove that limit of $2^n\sin(π/2^n)$ as $n$ approaches infinity is $π$. Generally I can tell limit of each term of product of $∞$ and $0$ respectively, but am little confused ...
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0answers
19 views

Problem with limits sinus in exp.

I have limits $ \lim_{x\to \infty} (3^{sinx \pi}-1).ln(x^2+2)$ I do with left bracket to $e^{sin x \pi ln 3}$ But i don't know next step. Anyone help?
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1answer
19 views

Finding a limit on multiple square roots in a row?

Here are basically my two problems, which I have the answer from WolframAlpha: $$ \lim_{n\to\infty}(1-\sqrt 2-\sqrt{n+1}+\sqrt{n+2})=1-\sqrt 2 $$ $$ \lim_{n\to\infty}(\sqrt n-2\sqrt{n+1}+\sqrt{n+2})=0 ...
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3answers
37 views

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is this done correctly?

$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is it allowed to split a multi-variable limit into its component variables as in the next step? $= ...
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1answer
22 views

Show that $f_{n}^2(x)$ does not converge in $D^1({\Omega})$

Let $$ f_n(x) = \left\{ \begin{array}{ll} n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\ 0 & \mbox{otherwise}.\end{array} \right. \ $$ I have to show that $\lim_{n \to ...
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2answers
45 views

Why is $ \lim_{x\to 25}(25−x)/(\sqrt x−5)= -10$? [on hold]

I know the answer is $-10$ but I don't know where the negative sign is coming from. This is what I ended up with. $$\frac{(x-25)(\sqrt{x}+5)}{x-25} = (1)\sqrt x+5 = 10 $$ ...
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0answers
15 views

How could I prove that equivalence on limits of sequences?

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ (these properties may be ...
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2answers
23 views

Calculating limit of a series of series

Given is the following series I want to calculate the limit for $n \to \infty$. I already recognised the geometric series $\sum_{i=2}^n a^{i} = \frac{1}{1-a}$ for $a=e^\rho$ (since rho is ...
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0answers
22 views

limit of $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$

Limit of equation: $\frac{(k+1)[x^k(a+sin(x^{-k-1}))-x^{-1}cos(x^{-k-1})]}{k[x^{k-1}(a+sin(x^{-k}))-x^{-1}cos(x^{-k})]}$ as $x \to 0$ and k=0,1,2,... My calculation steps: $=\frac{k+1}{k} ...
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5answers
134 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
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2answers
23 views

How do I use the ratio test to determine convergence or divergence in this problem?

I have the problem: $$a_{n} = \frac{e^{n+5}}{\sqrt{n+7}(n+3)!}$$ I am told to use the ratio test to determine convergence or divergence (or the test could be inconclusive). So I take the limit: ...
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1answer
16 views

Heine definition of limit of a function at infinity using sequences

I couldn't find the answer neither on Google, nor this website, so decided to ask. The Heine definition of limit: from Wikipedia $\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with ...
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2answers
45 views

$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$ iff $\lim_{n\to\infty}{(a_{n}-b_{n})=0}$ [on hold]

I need to proof or disproof that: $$\lim_{n\to\infty}{a_{n}} =\lim_{n\to\infty}{b_{n}}$$if and only if $$\lim_{n\to\infty}{(a_{n}-b_{n})=0}$$
3
votes
2answers
52 views

Proving limit of $|1-z|^2$ as $z \to i$ is 2

First off, apologies for my formatting. This is my first post and I'm still unfamiliar with MathJax and Latex, so I'm doing the best that I can. So I'm trying to prove that the limit of $|1-z|^2$ ...
1
vote
1answer
18 views

$a_n\geq b_n$ for $n>\bar{n}$ implies $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$

Consider two sequences of real numbers $\{a_n\}_n, \{b_n\}_n$. I know that if $a_n\geq b_n$ $\forall n$ then $\limsup_{n\rightarrow \infty}a_n\geq \limsup_{n\rightarrow \infty}b_n$. Suppose ...
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votes
4answers
64 views

find $\lim_{n\to\infty}(1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^4})\cdots(1+\frac{1}{3^{2^n}})$

$$\lim_{n\to\infty}\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^4}\right)\cdots\left(1+\frac{1}{3^{2^n}}\right)$$ You have to find the given limit when $n$ tends to ...
0
votes
1answer
39 views

Find the following limits without using l'Hopital's rule [on hold]

I really need your help on finding the following limits without using l'Hopital's rule: I think the second limit is zero, but I'm not sure if I have solved that correctly. Thanks in advance!
2
votes
2answers
47 views

Prove the following limit identity $\;\displaystyle\lim_{x\to\infty}\left( \csc\frac{m}{n+x}-\csc\frac{m}{x}\right)=\frac{n}{m}$

I am trying to prove the limit I came up with: $$\lim_{x\to\infty}\left(\csc\dfrac{m}{n+x}-\csc\dfrac{m}{x}\right)=\dfrac{n}{m}$$ This fact came from the double generalization of the special case ...
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3answers
78 views

Prove $\lim_{n \to \infty} \frac{\ln(n)}{n}=0$ without L'Hospital's Rule

Prove the following without using L'Hospital's Rule, integration or Taylor Series: $$\lim_{n \to \infty} \frac{\ln(n)}{n}=0 $$ I began by rewriting the expression as: $$\lim_{n \to ...
2
votes
2answers
56 views

Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$

Let $f:[0, +\infty) \rightarrow \mathbb{R} $ be a bounded function in each bounded interval. If $$\lim_{x \to +\infty} [f(x+1) - f(x)] = L$$ then $$\lim_{x \to +\infty} \frac{f(x)}{x} = L$$ I tried ...
0
votes
3answers
36 views

Arc length of a sequence of semicircles.

Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$. Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one ...