# Tagged Questions

Questions on the evaluation and properties of limits.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{x^2+3y^2}$?
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### 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. ...
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### 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'...
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### 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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### 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 ...
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### 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 $$\lim_{z\to 0}\frac{g(z)}{z}=2.$$ ...
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### 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 $x=0+h$ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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: $$\lim_{x\to4}(x-4)\cdot\cot(... 3answers 42 views ### Identity with exponential function: \lim_{n\to\infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2} Could you please explain me how we got this identity \lim_{n\rightarrow \infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2} when we know \lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e Thanks! 0answers 33 views ### Continous functions and zeros How to prove following theorem? If sequence \{f_n\} of continous real functions with domain D \subset \mathbb{R} is compact convergent to f and sequence \{x_n\} with D satisfies f_n(x_n) = ... 0answers 16 views ### Help with a confusing Multivariable limit. Hey guys writing my Second year maths exam tomorrow and upon going through some old exam questions I've come across one I'm having difficulty with. http://www5b.wolframalpha.com/Calculate/MSP/... 1answer 37 views ### How to solve: \ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n  How can I solve:$$\ \lim_{n \to +\infty} \frac{n^n + \frac {1}{n}}{(n + \frac {1}{n})^n} \ t^n $$tis a whole number. Thank you very much! Please tell me your ... 0answers 56 views ### Limit of \sum_{k=0}^{n}\frac{1}{2k+n} and similar Examine wether following sequences have limits and if yes - find them. a)\sum_{k=0}^{n}\frac{1}{2k+n} b)\sum_{k=0}^{n}\frac{(-1)^n}{2k+n} c)\sum_{k=0}^{n}\frac{(-1)^k}{2k+n}(\frac{1}{3})^k a)... 2answers 56 views ### When to rationalize to repair continuity, and why does it work? I was working on a question out a GRE math prep book: "Find the inverse of f(x) = \frac{x}{1-x^2} that works for all x \in \mathbb{R} where f is defined over (-1,1)" (works meaning is well ... 5answers 3k views ### A very curious rational fraction that converges. What is the value? Is there any closed form for the following limit? Define the sequence$$ \begin{cases} a_{n+1} = b_n+2a_n + 14\\ b_{n+1} = 9b_n+ 2a_n+70 \end{cases}$$with initial values a_0 = b_0 = 1. ... 2answers 50 views ### How do I solve \lim(1+1/x)^{x^2y/(x+y))} How do I solve this limit: it looks like euler can be used here, any ideas? The answer is 1. 3answers 80 views ### A basic question about limits [closed] How does one compute \lim\limits_{(x,y)\to (0,0)}\frac{2x^2 y}{x^4+y^2}? 3answers 73 views ### What is the limit for e^{\,x-1} as x tends to infinity? [closed] What is the limit$$\lim\limits_{x\rightarrow\infty} e^{\,x-1} ?$$Thanks. 3answers 138 views ### Prove \lim_{(x,y)\to (0,0)} \frac{x^2 y^3}{x^4 + y^4} =0 without \varepsilon - \delta. Unlike Multivariable Delta Epsilon Proof \lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4} --- looking for a hint I would like to avoid the \varepsilon - \delta criterium. Prove$$\lim_{(x,y)\to (0,0)...
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 ...
### 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.