# Tagged Questions

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### Evaluating a limit with variable in the exponent

For $$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$ I have to use the L'Hospital"s rule, right? So I get: $$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$ And ...
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### How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?

How do you find $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$ I know it's $-1$, but I had to plot it.
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### Prove that the limit $\displaystyle\lim_{x\to \infty} \dfrac{\log(x)}{x} = 0$

Prove $$\lim_{x\to \infty} \dfrac{\log(x)}{x} = 0$$ and $$\lim_{x\to \infty} \dfrac{\log(x)}{x^n} = 0$$ From the definition of $\log(x)$, $$\log(x) = \int_1^x \dfrac{1}{t} dt$$ Since $1$ ...
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### Simple inequality help

I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$\frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \$$ $$\lim_{k \rightarrow \infty} f(k)=0$$ ...
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### Limit with prime sequence and inverse logintegral

I found formula below$$\lim_{n\to\infty}\frac{\operatorname{li^{-1}}(n)}{p_n}=1$$ where $\operatorname{li^{-1}}(n)$ is inverse logintegral function and $p_n$ is prime number sequence. Can anyone ...
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### Find the limit of a sequence defined as solution to equation

We can easily prove that the equation of variable $x$ $$(E_{n}): \frac{x(\ln x)^n}{1+x}=\frac{e}{2(e+1)}$$ has a unique solution $u_{n}$ in $[1,e]$ for all integers $n$ greater than $1$. Let's call it ...
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### How can I calculate limit of division of two logarithms

How can I calculate $\lim\limits_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$. Where $a$ and $b$ are two integers.
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### Stuck on a homework question: if $t = \frac{1}{x}$

If $\displaystyle t = \frac{1}{x}$ then a) Explain why $\displaystyle\lim_{x \to 0^-}f(x)$ is equivalent to $\displaystyle\lim_{t \to -\infty}f\left(\frac{1}{t}\right)$ b) Using that, ...
### What is the limit of multiple logarithm quotient $\frac{\log \log n}{\log n}$
Could somebody check if this is correct? $$\lim_{n \to \infty} \frac {\log_{2}(\log_{2}(n))}{\log_{2}(n)}$$ I exponantiate the numerator and the denominator with 2 \frac ...