Tagged Questions
1
vote
3answers
46 views
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 ...
4
votes
4answers
103 views
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.
3
votes
3answers
50 views
If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
We are dealing with $z \in \mathbb{C}$.
I know that
$$
\left(1+ \frac{z}{n} \right)^n \to e^{z}
$$
as $n \to \infty$. So intuitively if $z_n \to z$ then we should have
$$
\left(1+ \frac{z_n}{n} ...
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
1
vote
2answers
48 views
Limit, log rule
I have couple of question on this part of equation -
\begin{align}
\lim_{n\to \infty } \frac{ 7 \cdot \sqrt{n}}{\log(n)}- \lim_{n\to \infty} \frac{1}{n\cdot \log(n)} &=\lim_{n\to \infty} \frac{7 ...
6
votes
1answer
69 views
What is $ \lim_{x \to 0} \log_0(x) $?
As per the title; what is $ \lim_{x \to 0} \log_0(x) $ ?
According to WolframAlpha: $$ \lim_{x \to 0} \log_0(x) = 0 $$ but how is this possible?
Surely the limit should be indeterminate since ...
1
vote
2answers
136 views
Natural logarithm limit
Is
$$\lim_{n\rightarrow +\infty}\ln\left(\frac{n+1}{n}\right)=0?$$
Because it is $\ln(1+\frac{1}{n})$ and $\frac{1}{n}$ tends to $0$, since $n$ tends to infinity, so the limit becomes ...
2
votes
2answers
85 views
Find the limit of the sequence containing logarithm??
Find $\lim_{n→∞} [log(2+3^n)]/2n$
I have my work till the very last step then i dont know how to continue
$\lim_{n→∞} [log(2+3^n)]/2n$
=$\lim_{n→∞} log(3^n)+\lim_{n→∞} log[(2+3^n)/3n]$
...
3
votes
2answers
100 views
Limit of logarithms without l'Hospital
This is my first post so I hope you forgive any formatting mistakes. This is a task out of a training exam, I may add that we have not yet introduced l'Hospital or derivatives. We have to determine ...
3
votes
3answers
82 views
How can I show that $f(x) = (x^2)/(1-e^x)$ has global minimum at $(0, +\infty)$?
I showed that $\lim f(x) = 0$ at both the $0$ end and $+\infty$ end.
What is the proper way to finish the proof?
0
votes
1answer
68 views
Is $\lim_{x \to x_0} \log(f(x)) = \log\lim_{x \to x_0} f(x)$ always true?
This property is always true? If yes I would like a proof, otherwise an counterexample.
$$\lim\limits_{x \to x_0} \log(f(x)) = \log\lim\limits_{x \to x_0} f(x)$$
4
votes
2answers
121 views
Logarithm as limit
Wolfram's website lists this as a limit representation of the natural log:
$$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$
Is there a quick proof of this? Thanks
1
vote
5answers
281 views
L'Hospital's Rule Question.
show that if $x $ is an element of $\mathbb R$ then $$\lim_{n\to\infty} \left(1 + \frac xn\right)^n = e^x $$
(HINT: Take logs and use L'Hospital's Rule)
i'm not too sure how to go about answer this ...
3
votes
2answers
105 views
Evaluate: $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$
Evaluate: $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$$
attemp: Take $P=\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$ . Then taking log both side .$$\ln ...
0
votes
4answers
93 views
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 $1$ ...
2
votes
1answer
64 views
Simple inequality help
I need a function $f(x)$ that satisfies the properties bellow for all integers $k$
$$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $$
$$ \lim_{k \rightarrow \infty} f(k)=0 $$
...
1
vote
1answer
39 views
Does the logistic function uniquely satisfy these three conditions?
Given
$$r(t)=\frac{f(t)}{1-F(t)} \tag{Eq. 1}$$
where
$$f(t)=\frac{dF}{dt} \tag{Eq. 2}$$
and the conditions:
$$\lim_{t\rightarrow \infty} r(t)=1 \tag{Eq. 3}$$
$$\lim_{t\rightarrow \infty} F(t)=1 ...
-2
votes
2answers
115 views
Proving the limit of $\frac{\log(n)^{\log(n)}}{1.01^{n}}$
Can anyone please show me a simple way (if there is one) to show that
$$\lim_{n\to \infty}\frac{\log(n)^{\log(n)}}{1.01^{n}}=0$$
And that
$$\lim_{n\to \infty}\frac{1.01^{n}}{n!}=0$$
I've checked that ...
9
votes
7answers
391 views
Proving limit with $\log(n!)$
I am trying to calculate the following limits, but I don't know how:
$$\lim_{n\to\infty}\frac{3\cdot \sqrt{n}}{\log(n!)}$$
And the second one is
$$\lim_{n\to\infty}\frac{\log(n!)}{\log(n)^{\log(n)}}$$
...
1
vote
2answers
62 views
How to reduce the limit one gets when deriving the derivative of the general exponential function?
When applying the definition of a derivative to $\frac{d}{dx}b^x$ and a little algebra one arrives to
$$b^x\times\lim\limits_{h \to 0}\frac{b^h - 1}{h}$$
where of course that limit equals $\ln(b)$. ...
1
vote
2answers
119 views
Interchanging limits and logarithms
This is probably not too smart, just wondering of the name of this rule:
$$
\log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x)
$$
A reference to a source and/or proof would be good too.
1
vote
1answer
59 views
Find $\lim_{x\to \infty} \ln(\exp(\operatorname{LmW}(x))+1)(\exp(\operatorname{LmW}(x))+1) - x - \ln(x)$
Find $\lim_{x\to \infty} \ln(e^{\operatorname{LambertW}(x)}+1)(e^{\operatorname{LambertW}(x)}+1) - x - \ln(x)$
Where the $LambertW$ function is defined here : http://en.wikipedia.org/wiki/Lambert_W
...
2
votes
2answers
432 views
limits of logarithm
I am trying to understand the definition of a logarithm, because when I was trying to find the derivative of $2^x$ I got $$2^x \lim_{h \to 0} \frac{2^h-1}{h}$$ which I have found by searching to be ...
1
vote
5answers
146 views
Limit of a recursively defined bivariate function.
Let m and n be positive integers.
Let $f(m,0)=m$
Let $f(m,n)= e \ln(f(m,n-1))$
$$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$
Where $C$ is a constant.
It seems ...
0
votes
2answers
152 views
Find the limit of $f(x)$ involving a sum of logarithms.
I need to find
$\lim_{x\rightarrow0}f(x)$ for the following function:
$f:(0,+\infty)$
$f(x)=[1+\ln(1+x)+\ln(1+2x)+\dots+\ln(1+nx)]^\frac{1}{x}$
I tried writing the logarithms as products:
...
1
vote
6answers
145 views
How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$
How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$
1
vote
4answers
155 views
How can I calculate $\lim_{x \to 0} \log(\cos(x))/\log(\cos(3x))$ without l'Hopital?
How can I calculate the following limit without using, as Wolfram Alpha does, without using l'Hôpital?
$$
\lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x}
$$
2
votes
3answers
55 views
Big-$\mathcal{O}$ bounding of sums of logarithmic functions
I am reading a text which states that $$\sum \limits_{n \leq X} \left(\log X - \log n \right) = \mathcal{O}(X)$$ I can't quite see why this is true, though I can certainly believe it. Could anyone ...
3
votes
3answers
206 views
How to find the limit $\lim \limits_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]$
Solve $\space \begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]
\end{align*}$.
I did this way:
$$\begin{align*}
\lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] ...
3
votes
2answers
90 views
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 ...
2
votes
1answer
123 views
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 ...
0
votes
2answers
90 views
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.
1
vote
2answers
113 views
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, ...
2
votes
3answers
737 views
Can someone explain log?
I have done some higher math, being an engineer. They usually focus on getting the correct answer as supposed to actually understanding math. Glorified calculators, basically. I know how to do stuff ...
1
vote
3answers
306 views
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 ...
