Yuck. So I have no chance to find out what is allowed if someone asks "Find the limit of xyz using only notable limits (no Taylor expansion nor L'Hospital)".
Hey, this is a question that a friend of mine asked a while ago... Although I read the hints given to him, I barely understood most of it. Would really love some help if someone can assist . math.stackexchange.com/questions/568179/…
@leo we might like to write the limit as $$\lim_{x\to0}\left( \left(\frac{1}{\log({x+1})}-\frac{1}{x}\right)-\left(\frac{1}{\log({x+\sqrt{x^2+1}})}-\frac{1}{x}\right)\right)$$
Here is a limit that can be computed directly by performing the integration and then taking
the limit, but the way is rather ugly. What else can we do? Might we avoid the integration?
$$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$$
@leo kids with no knowledge in calculus modified my questions the way they wanted. Then, when I said I don't agree with that, I was warned by a moderator.
@leo never, but never in this life I'll be active on main.
relevant question: Integral involving logarithm and cosine
What is a interesting definite integral function that is equal to the minium of three functions?
@Chris'ssis why do you think that says you must leave? People were simply mentioning that signatures should not be added to posts and that posts may be edited by others? That does not indicate "go away" to me.
relevant question: Integral involving logarithm and cosine
What is a definite integral function that is equal to the minium of the set of any three functions?
I have seen that a definite integral could be represent in two differnet forms in the relevant question, i wonder if a definite integral function could be represent as three different forms of a function @ robjohn
@robjohn That's really great! I've never seen that proof before although I also thought to express things in terms of an infinite series but I didn't do it yet. I upvoted that.
@robjohn I've found an elementary proof in one line. :-))))))))
We will use that $\lim\limits_{x\to0}\frac{\sin(x)}{x}=1$, which can be shown geometrically.
First note that
$$
\begin{align}
\frac1{\vphantom{()^2}x^2}-\frac1{\sin^2(x)}
&=\frac{\sin^2(x)-x^2}{x^2\sin^2(x)}\\
&=\frac{\sin^2(x)-x^2}{x^4}\left(\frac{\sin(x)}{x}\right)^{-2}\\
&=\frac{\sin(x)-x}{x^...
i have this theorem and a part of the proof
i don't understand why they take an orthogonal projection and how they use the implicite function theorem ??
Please
Thank you
@PedroTamaroff Jesus, that was a brilliant shot! :D My nicest limit proof in the last period of time. (last month since I also had an amazing proof for another limit some weeks ago)
@PedroTamaroff let me show you the limit (it's posted here on MSE)
Say $\log x + \log(x + 2) = \log 3$. I would solve this extremely simple equating by noting that $\log 3 = \log 1 + \log( 1 + 2)$ hence $x=1$. And this is correct.
But I have a hard time explaining why it is correct. By doing it the proper way, the left handside turns into a quadratic equating giving two solutions, where the one is thrown away as it is negative.
@robjohn from that limit we can easily elementarily prove $\lim_{x\to0} \frac{x-\sin(x)}{x^3}=\frac{1}{6} $. And from this one, many other problematic limits where only elementary tools are allowed.
Here is the beauty $$\lim_{n\to\infty} \frac{\displaystyle\int_0^1 \left(\frac{\log(2+x)}{2+x}\right)^{n} \ dx}{\displaystyle\int_0^1 \left(\frac{\log(2+x)}{2+x}\right)^{n+1} \ dx}$$
@EnjoysMath I'm out of practice with abstract algebra. By the way, I don't push you to solve anything, you may simply ignore my posts if you don't like them.
@robjohn is such a limit studied in precalculus? $\lim_{n\to\infty} \frac{\displaystyle\int_0^1 \left(\frac{\log(2+x)}{2+x}\right)^{n} \ dx}{\displaystyle\int_0^1 \left(\frac{\log(2+x)}{2+x}\right)^{n+1} \ dx}$
Let $x$ be a twice differentiable equation. Suppose that the function $x$ satisfies the following integral equation:
$x(t)$ + $\displaystyle\int_{0}^{t} \frac{x(s)}{s + 1}\ ds= t+2$
Find the function $x$.
I tried finding the integral above by using integration by parts and differentiating $x(s...
You wouldn't say that linear equations are not high school algebra just because a high school algebra student couldn't do $a\int_0^xe^{-t^2}\,\mathrm{d}t-bx=0$
Suppose $K$ is separated, as in math.stackexchange.com/questions/387555/…. Then it can be written as the union of two separated sets $A, B$. Therefore $K \subset A \cup B$.
Now if both $K \subset A$ and $K \subset B$ then we have $A \cap B \neq \emptyset$ a contradiction.
Here is a limit that can be computed directly by performing the integration and then taking
the limit, but the way is rather ugly. What else can we do? Might we avoid the integration?
$$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$$
relevant question: Integral involving logarithm and cosine
What is a interesting definite integral function that is equal to the minium of three functions?
i have this theorem and a part of the proof
i don't understand why they take an orthogonal projection and how they use the implicite function theorem ??
Please
Thank you
Let $x$ be a twice differentiable equation. Suppose that the function $x$ satisfies the following integral equation:
$x(t)$ + $\displaystyle\int_{0}^{t} \frac{x(s)}{s + 1}\ ds= t+2$
Find the function $x$.
I tried finding the integral above by using integration by parts and differentiating $x(s...
