Questions on the evaluation and properties of limits.

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3answers
51 views

Limit of fibonacci sequence

Let $f_n$ be the $n$th Fibonacci number. Find constants $a$ and $b$ such that $$\lim_{n\to\infty} \frac{f_n}{a\cdot b^n} = 1$$ I'm somewhat confused on how to approach this problem. I know the ...
0
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0answers
9 views

Vector elements converging to the same value - a proof by contradiction

Note: I'm going to simplify the proposition and proof in this question a bit to avoid a large number of definitions and theorems - hopefully I don't remove anything vital. I'm afraid the material here ...
3
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4answers
34 views

Find $\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$

I have to solve this limit: $$\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$$ I know the answer is $-\infty$. My question is, can I do this: $$\ln[\lim_{n\to\infty}n^{n}\cdot(n+1)^{-n-1}]$$ If not, how ...
-4
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2answers
39 views

How to show that $\lim_{(x,y)\rightarrow (0,0)} 2^{\frac{xy}{x^2+3y^2}}$ does not exist? [on hold]

How to show that $\displaystyle\lim_{(x,y)\rightarrow (0,0)} 2^{\frac{xy}{x^2+3y^2}}$ does not exist? In other words, how can I solve this: $\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{x^2+3y^2}$?
0
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5answers
56 views

Limit of this sequence [duplicate]

So guys I need to find the limit of: $\displaystyle\lim_{n \to \infty}\left(\sqrt{n^2+2n+5}-n\right)$ The quadratic equation is hard to factorise and I really struggle to answer these questions. ...
0
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1answer
18 views

Prove that $\lim_{x\rightarrow 1} \frac{\int_0^xg(t)dt-\int_0^1g(t)dt-\int_0^1f(t)dt(x-1)}{(x-1)^2=\frac{f(1}{2}}$

Prove that $$\lim_{x\rightarrow 1} \frac{\int_0^xg(t)dt-\int_0^1g(t)dt-\int_0^1f(t)dt(x-1)}{(x-1)^2}=\frac{f(1}{2}$$ Now I now this is a limit of the form $\frac{"0"}{"0"}$ which means I can use L'...
0
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2answers
47 views

How can we say the derivative is exact if the difference quotient has a domain restriction?

I think I've finally been able to voice my confusion when it comes to derivatives and limits. Let's first look at the difference quotient for a function $f(x)=x^2$ $$\lim_{h\to0} \frac{f(x+h)-f(x)}{...
2
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1answer
49 views

How do I show that $E-\gamma=\lim_{j\to \infty}\sum_{n=1}^{j}n\left({1\over 2^n-1}-{1\over 2^n}+\cdots-{1\over 2^{n+1}-2}\right)$

Given the Erdos-Borwein's constant $E=\sum_{n\geq 1}\frac{1}{2^n-1}$ and the Euler-Mascheroni constant $\gamma=0.5772156...=\sum_{n\geq 1}\left[\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right]$ how ...
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3answers
36 views

Solving limit $x$ tending to $0$ for $(\tan(x) - \sin(x))/x^3$

I have a solution which gives the answer as $1$: Write $\tan(x)$ as $\sin(x)/\cos(x)$. Take $\sin(x)$ common, and multiply&divide by $\cos(x)$. Rewrite $(1-\cos(x))$ as $2\sin^2(x/2)$ Apply ...
3
votes
1answer
49 views

Study of differentiablity of function

Study the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ $f(x,y)=\begin{cases} \frac{x^3+y^3}{x^2+\left|y\right|} & (x,y)\ne(0,0) \\ 0 &(x,y)=(0,0) \\ ...
2
votes
2answers
43 views

Intuition behind LHS of squeeze theorem

I am reading the solution to the following problem: Evaluate the limit of $$\lim_{x \rightarrow \infty } \left(\int^{\pi / 6}_0 (\sin t)^x dt \right)^{1/x}$$ The first step was stating that for $t \...
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0answers
15 views

Finding a constant using limits and a given piece wise function.

The question states Find $\theta$ which makes $f(x)$ is continuous everywhere. Sorry about the notation , I don't have enough time to lookup the piece wise function syntax and I can't recall it :-) $...
0
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1answer
30 views

Question involving continuity of function

Problem: Function $f$ is defined: $f(x)=x^2$ for $x\in \mathbb Q$ and $f(x)=x$ for irrational $x$. I have to check continuity of function. My work: Let $c\in \mathbb R\setminus \mathbb Q$. ...
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5answers
50 views

Compute $\lim_{x\to 0}\tan (\pi/4+x)^{1/x}$ [on hold]

$$\lim_{x\to 0}\tan (\pi/4+x)^{1/x}$$ How to solve this problem?
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0answers
39 views

How is the derivative exact if it's the standard part? [on hold]

I understand that no such infinitesimal value exists in our universe, so we round to the nearest real value when taking the derivative. Such as how $2x+\Delta x$ equals $2x$ when we take the standard ...
1
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0answers
31 views

Clarification about consequence of physics' first principle of thermodynamics

I'm reading the book Physics, by Tipler, and I'm confused at the following statement: [...] From the first principle of thermodynamics, $\Delta U = Q+ W$. Suppose an ideal gas is given heat while ...
0
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1answer
34 views

Limits involving $e$ [on hold]

I am looking to solve the following limits. $\displaystyle\lim_{x\to-\infty}5e^{-x}$ and $\displaystyle\lim_{x\to2}\frac{1}{2e-ex}$ Any help would be appreciated.
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2answers
47 views

Help with limit proofs [on hold]

I'm having trouble finding the following limit. Any help for a different approach would be great. Thank you!
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3answers
74 views

Question involving limit

How to find the following limit: $$\lim_\limits{n\to\infty} (ne\sqrt[n]{\ln{(1+e^n)}-n}-n)$$ Thanks in advance!
3
votes
3answers
80 views

Limit of $\frac{\pi^h-1}{h}$ as h approaches zero

Can someone help me find this limit here. I only know how to use L'Hospital's rule but I want to be able to evaluate this limit without using differentiation. $$\lim \limits_{h \to 0} \frac{\pi^h-1}{h}...
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5answers
46 views

Integration with limits and options.

I found this exercise in an old exam but I don't know how to attack it because is a limit of an integration and I don't know if the limit affects the process of the integral or it makes it easier. The ...
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3answers
49 views

Find $\lim\limits_{x \to \infty} \frac{x}{(\log{x})^n}$

Find $\displaystyle \lim_{x \to \infty} \dfrac{x}{(\log{x})^n}$. My book says that $\displaystyle \lim_{x \to \infty} \dfrac{(\log{x})^n}{x} = \lim_{y \to \infty} \dfrac{y^n}{e^y} = 0$, but I don't ...
2
votes
1answer
31 views

Evaluation of $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$

Find the value of $$\bigg[ \lim_{x\to 0} \frac{e-(1+x)^{1/x}}{\tan x}\bigg]$$ where $[.]$ represents floor function (or greatest integer function) I wrote it as $\bigg[ \lim_{x\to 0} \frac{e-(1+x)^...
3
votes
1answer
39 views

Is the following integral identity true or not? [on hold]

Is the following statement true or not?$$\int_{-\infty}^\infty xf(x)\,dx = \left. {d\over{dt}} \int_{-\infty}^\infty e^{tx}f(x)\,dx\right|_{t = 0}$$
2
votes
2answers
62 views

Is $1/(x^2 + y^2)$ continuous?

I'm trying to check whether the function $$ f(x,y)=\begin{cases} \dfrac{1}{x^2+y^2} & \text{for $(x,y)\ne(0,0)$}\\[6px] 0 & \text{for $(x,y)=(0,0)$} \end{cases} $$ is continuous. My problem is ...
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0answers
26 views

The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows \begin{equation}\label{cp26} F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...
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votes
2answers
71 views

Is f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}} $with f(0,0)=0 continuous in (0,0) [duplicate]

I believe that the function: f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}}$ is continuous on the point (0,0) but i can't prove it. I know you have to choose something like $x=cy^{2}$(with c a constant) to prove ...
5
votes
2answers
279 views

Fallacy limit problem - Where is the mistake?

This problem comes from our text book. Evaluate $$ \lim\limits_{x \to 0}\frac{2^x-1-x}{x^2} $$ without using either L'Hopital's rule or Taylor series. The picture below shows the solution given by ...
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0answers
19 views

Am I understanding this limit correctly?

Given that $\Phi$ and $y'$ are continuous functions. If $$T_n =\frac{y(x_{n+1}) - y(x_n)}{h}-\Phi(x_n,y(x_n);h)$$ and $h \rightarrow 0, n \rightarrow \infty$ with $\lim_{n \rightarrow \infty}x_n = x ...
2
votes
2answers
43 views

Find $\lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right)$ where $\lim_{z\to 0}\frac{g(z)}{z}=2.$

This limit seems different to me than all the other multi variable limits already asked on this site. Let $g \colon \mathbb R \to \mathbb R $ be such that $$ \lim_{z\to 0}\frac{g(z)}{z}=2. $$ ...
4
votes
3answers
74 views

Dilemma about value of limit

$$ \lim_{x\to 0^{+}}\left[\left(1+\frac{1}{x}\right)^x+\left(\frac{1}{x}\right)^x+\left(\tan(x)\right)^{\frac{1}{x}}\right]$$ Attempt: I used $\tan(x)\approx x$ also $(1+n)^{1/n}=e$ so I let $x=0+h$ ...
2
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1answer
66 views

Is a limit a formalized infinitesimal?

From what I understand after thinking about this, delta epsilon really seems to formalize the notion of an infinitesimal. The constraint $0<|\delta-c|$ combined with the fact that there is no real ...
3
votes
2answers
53 views

limit of an integral over a function

Calculate $\displaystyle\lim_{x\to0}{F(x)\over g(x)}$, where $ g(x)=x$ and $\displaystyle F(x)=\int_0^x {e^{2t}-2e^t+1\over 2\cos3t-2\cos2t+\cos t} \, dt$. i'd love for someone to explain not only ...
5
votes
2answers
93 views

Finding Limit of Nested/Continued Logarithm

For a sequence $a_n$ defined by: $$a_1 = \ln(1)$$ $$a_2 = \ln\left(\frac{1}{\ln(2)}+1\right)$$ $$\dots a_n = \ln\left(\frac{1}{\ln(\frac{1}{\ln(\dots 1/\ln(n ))}+1)}+1 \right)$$ with $n$ ...
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1answer
36 views

Finding the limit of these functions

Do you mind explaining me how to find the limit of these functions? $\lim_{n\rightarrow \infty}\frac{7n^5-2}{(n+4)^5n}$ $\lim_{n\rightarrow \infty}\frac{(n^3+1)n^3}{((n+1)^3+1)(n+1)^3}$ Thank you ...
1
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1answer
34 views

How to calculate $\lim_{x\to4}(x-4)\cdot\cot(x-4)$

It's been a while since I've done calc, so I'm trying to review by reading "Calculus Demystified" by Steven G. Krantz. Question 1c at the end of chapter 2 has me stumped: $$\lim_{x\to4}(x-4)\cdot\cot(...
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votes
3answers
42 views

Identity with exponential function: $\lim_{n\to\infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$

Could you please explain me how we got this identity $\lim_{n\rightarrow \infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$ when we know $\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e$ Thanks!
0
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0answers
33 views

Continous functions and zeros

How to prove following theorem? If sequence $\{f_n\}$ of continous real functions with domain $D \subset \mathbb{R}$ is compact convergent to $f$ and sequence $\{x_n\}$ with $D$ satisfies $f_n(x_n) = ...
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0answers
16 views

Help with a confusing Multivariable limit.

Hey guys writing my Second year maths exam tomorrow and upon going through some old exam questions I've come across one I'm having difficulty with. http://www5b.wolframalpha.com/Calculate/MSP/...
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1answer
37 views

How to solve: $\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $

How can I solve: $$\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $$ tis a whole number. Thank you very much! Please tell me your ...
1
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0answers
56 views

Limit of $\sum_{k=0}^{n}\frac{1}{2k+n}$ and similar

Examine wether following sequences have limits and if yes - find them. a)$\sum_{k=0}^{n}\frac{1}{2k+n}$ b)$\sum_{k=0}^{n}\frac{(-1)^n}{2k+n}$ c)$\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k$ a)...
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vote
2answers
56 views

When to rationalize to repair continuity, and why does it work?

I was working on a question out a GRE math prep book: "Find the inverse of $f(x) = \frac{x}{1-x^2}$ that works for all $x \in \mathbb{R}$ where $f$ is defined over $(-1,1)$" (works meaning is well ...
16
votes
5answers
3k views

A very curious rational fraction that converges. What is the value?

Is there any closed form for the following limit? Define the sequence $$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$ with initial values $a_0 = b_0 = 1$. ...
1
vote
2answers
50 views

How do I solve $\lim(1+1/x)^{x^2y/(x+y))}$

How do I solve this limit: it looks like euler can be used here, any ideas? The answer is 1.
0
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3answers
80 views

A basic question about limits [closed]

How does one compute $\lim\limits_{(x,y)\to (0,0)}\frac{2x^2 y}{x^4+y^2}$?
0
votes
3answers
73 views

What is the limit for $e^{\,x-1}$ as $x$ tends to infinity? [closed]

What is the limit $$\lim\limits_{x\rightarrow\infty} e^{\,x-1} ?$$ Thanks.
3
votes
3answers
138 views

Prove $\lim_{(x,y)\to (0,0)} \frac{x^2 y^3}{x^4 + y^4} =0$ without $\varepsilon - \delta$.

Unlike Multivariable Delta Epsilon Proof $\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4}$ --- looking for a hint I would like to avoid the $\varepsilon - \delta$ criterium. Prove $$\lim_{(x,y)\to (0,0)...
2
votes
2answers
78 views

Trigonometric and exp limit

Evaluation of $$\lim_{x\rightarrow \frac{\pi}{2}}\frac{\sin x-(\sin x)^{\sin x}}{1-\sin x+\ln (\sin x)}$$ Without Using L hopital Rule and series expansion. $\bf{My\; Try::}$ I have solved it ...
1
vote
1answer
47 views

Limit of the fraction $\dfrac{n(n+1)^\alpha}{\sum_{k=1}^nk^\alpha}$

I'm stuck in calculating the following limit: $$L=\displaystyle\lim_{n\to\infty}\dfrac{n(n+1)^\alpha}{\sum_{k=1}^nk^\alpha}$$ For what values of $\alpha\in\mathbb{R}$ $L$ has a finite value? Thanks.