All Questions
Tagged with logarithms limits
633
questions
4
votes
1
answer
178
views
Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
- 936
1
vote
2
answers
77
views
How to calculate the limit $\lim_{x \to 0}\frac{\ln(1 + \sin(12x))}{\ln(1+\sin(6x))}$ without L'Hôpital's rule?
$$
\lim_{x \to 0}\frac{\ln(1+\sin12x)}{\ln(1+\sin6x)}
$$
I know it's possible to calculate this limit just by transforming it; I think you need to use the knowledge that
$$
\lim_{x \to 0}\frac{\ln(1+x)...
1
vote
2
answers
59
views
Unexpected asymptotic logarithm behavior
I have recently seen a rather confusing asymptotic property of logarithms:
$$
\log(n^4 + n^3 + n^2) \leq O(\log(n^3 + n^2 + n))
$$
I find this very unintuitive. Why would the log of a bigger ...
-1
votes
4
answers
97
views
Calculate $\lim_{n\to \infty}\int_n^{n+1}\frac{1-\ln{x}}xdx$ [duplicate]
$$
\mbox{Given}\ \operatorname{F}\left(x\right) \equiv
\int_{1}^{x}\frac{1-\ln\left(t\right)}{t}\,{\rm d}t,\ \mbox{calculate}\ \lim_{n \to \infty}\left[\operatorname{F}\left(n + 1\right) - \...
- 345
1
vote
1
answer
110
views
What is $\log_2{\aleph_0}$?
I understand that $\aleph_0$ is the cardinality of the natural numbers, as well as any set A, for which there’s a way to both match every element to of A to the natural numbers, and match every ...
- 205
1
vote
1
answer
59
views
Convergence for a sequence using logarithmic properties?
Given the sequence $x_n=\frac{\log(3n+2)}{\log(n^2+2)}$, which has the limit $\frac{1}{2}$ when n goes to infinity, give a formal proof of the limit using the epsilon definiton.
$$| x_n - L |< \...
- 41
0
votes
1
answer
60
views
What is the nature of the pole of the derivative of $\frac{1}{1- \ln(x)}$ at $x=0$?
I'm interested in the function $\frac{1}{1-\ln(x)}$ on positive real line.
One can experimentally see that
$$ \lim_{x \rightarrow 0^+} \left[ x \frac{d}{dx} \left[ \frac{1}{1 - \ln(x)} \right] \right] ...
- 17.5k
1
vote
2
answers
71
views
Is $y = (\log x)^u$ faster than $y = x$ for any $u$?
If $(\log x)^u\over x$ converges to a constant as $x \rightarrow \infty$, then the set $S$ of possible values of $u$ is
(A)$[-1,1]$
(B)$(-\infty,1]$
(C)$(-\infty,\infty)$
(D)None of the above
Here's ...
0
votes
2
answers
60
views
How to simplify complex logs
I have f(x) = (x+$\sqrt{x}$)$\log_2x$ and g(x) = x$log_2(x+\sqrt{x})$. How would I go about simplifying them and obtaining the limit at infinity of f(x)/g(x). So far, the best I have gotten for f(x) ...
- 39
0
votes
2
answers
40
views
Limit value of logarithm term using sum-representation [duplicate]
I want to correctly prove the following statement:
\begin{equation}
\lim_{x \rightarrow \infty} \frac{1}{x \cdot \ln\left(\frac{x + a}{x - a}\right)} = \frac{1}{2a} \ , \ a \in \mathbb{R}^{\ast+}
\...
1
vote
2
answers
72
views
Limit of $\frac{\log(R)^p}{ R^n }$ as $ R \rightarrow \infty$
I have been reading this old post:
Showing that an entire function with the following inequality is constant.
In the solution to the problem, we claim the following for $C$ constant, $n$ is a positive ...
- 19
0
votes
1
answer
66
views
Using standard limits while adding two functions
When is it allowed to substitute the value of a standard limit such as $\lim_{x\rightarrow0} (\sin x)/x=1$ and $\lim_{x\rightarrow0} \ln(1+x)/x=1$ while adding different functions?
For example if we ...
- 69
2
votes
1
answer
157
views
Iterating $\log(x\log(x\log(...)))$
For a real positive $x$, let $F(x)$ denote the sequence
$$
\left(\log x,\log(x\log(x)),\log(x\log(x\log(x))),\log(x\log(x\log(x\log(x)))),...\right),
$$
stopping at the first nonpositive value or ...
- 1,673
2
votes
0
answers
69
views
Show that $ \lim_{x \to -\infty} (1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n ) = 0$
Let $x$ be real and define the entire function $f(x)$ as
$$ f(x) = 1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n $$
Now we have that
$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (1 + \sum_{...
- 16.4k
2
votes
0
answers
65
views
Compare the growth rate of functions $f(n)=(2^{n+1}-1)^{1/\log n}$ and $n^{\log n}$
How do I compare the growth rate of functions $f(n)=(2^{n+1}-1)^{1/\log n}$ and $g(n)=n^{\log n}$
My Attempt
\begin{align}
\lim_{n\to\infty}\frac{\log f(n)}{\log g(n)}&=\lim_{n\to\infty}\frac{\log\...
- 7,674