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• Today, 10:37
I substituted $n=\frac 1{x^2}$, which also means that $x^2=\frac 1 n$. So we get $\Big(1 + \frac 32 x^2\Big)^{1/x^2} = \Big(1 + \frac 32 \cdot \frac... 7 replies | 76 view(s) • Today, 09:14 It's like this: $$\Big(1-\frac 12 x^2 + \ldots\Big)\Big(1+\frac 12 (2x)^2 - \ldots\Big)\\ =1\cdot 1 -\frac 12 x^2 \cdot 1+1\cdot \frac 12 (2x)^2 -... 7 replies | 76 view(s) • Today, 07:04 The series expansion of \cos x = 1 - \frac 12x^2 + \ldots. And the expansion of \frac 1{1-x} = 1+x+x^2+\ldots So:$$\frac{\cos(x)}{\cos(2x)}... 7 replies | 76 view(s) • Today, 01:08 Here is this week's POTW: ----- Suppose that the positive integers$x, y$satisfy$2x^2+x=3y^2+y$. Show that$x-y, 2x+2y+1, 3x+3y+1$are all... 0 replies | 33 view(s) • Today, 01:02 No one answered last week's POTW.(Sadface) Below is a suggested solution: The given function is the square of the distance between a point of the... 1 replies | 128 view(s) • Yesterday, 04:14 This is actually a proof of (ii)$\Rightarrow$(iii) in Lemma 3.2. So we are assuming that (ii) holds. In particular, since$\overline{ f(A) }$is... 1 replies | 55 view(s) • April 3rd, 2020, 22:31 I am reading Aisling McCluskey and Brian McMaster: Undergraduate Topology, Oxford University Press, 2014... ... and am currently focused on Chapter... 1 replies | 55 view(s) • April 2nd, 2020, 12:25 Klaas van Aarsen replied to a thread f in Calculus It appears the answer is: dkm Acronym for "don't kill me". Often used when somebody says something that somebody else finds really funny,... 5 replies | 158 view(s) • April 2nd, 2020, 11:56 topsquark replied to a thread f in Calculus I believe the proper question is "What the 'f'?" -Dan 5 replies | 158 view(s) • April 1st, 2020, 06:35 Not quite, although you started correctly. The limit in this case is as$x\to\infty$, so you want to see what happens when$x$gets large. This means... 3 replies | 144 view(s) • March 31st, 2020, 17:53 topsquark replied to a thread [SOLVED] 299What is the acceleration in Calculus Wait! What do you mean by you changed it to v(t)? The particle's position is x(t) = sin(t) - cos(t) means that v = \dfrac{dx}{dt}. You can't just... 4 replies | 125 view(s) • March 31st, 2020, 14:52 Hi Goody, and welcome to MHB! To prove that \lim_{x\to\infty}\frac{x-1}{x+2} = 1, you have to show that, given$\varepsilon > 0$, you can find$N$... 3 replies | 144 view(s) • March 30th, 2020, 05:31 Let's take a look at a couple of examples. If$f(x)=x$, then$f'(x)=1$and$f''(x)=0$. So$\lim\limits_{x\to +\infty}f'(x)\ne 0$, isn't it?... 21 replies | 404 view(s) • March 30th, 2020, 04:26 Yep. (Nod) And no, nothing more specific. 21 replies | 404 view(s) • March 30th, 2020, 03:32 I believe so yes. Consider$f(x)=\ell$. It satisfies all conditions, doesn't it? (Wondering) And it is monotone instead of strictly monotone. To... 21 replies | 404 view(s) • March 29th, 2020, 14:38 Indeed. (Thinking) 21 replies | 404 view(s) • March 29th, 2020, 14:26 Yep. (Nod) 21 replies | 404 view(s) • March 29th, 2020, 13:57 Then the inequality also holds yes. What if fill in, say,$f'(y)=-1$in the inequality? Would it satisfy it? (Wondering) 21 replies | 404 view(s) • March 29th, 2020, 13:19 That is a possibility yes. What happens if$f'(y)$is negative? (Wondering) 21 replies | 404 view(s) • March 29th, 2020, 13:08 We have an expression with$f(y)$,$f'(y)$, and$\ell$. And we already know that$\ell\in\mathbb R$, don't we? So it can't be$\pm\infty$either. ... 21 replies | 404 view(s) • March 29th, 2020, 10:27 Ah okay. But that is not the case now is it? (Wondering) Sounds like a plan. (Nod) 21 replies | 404 view(s) • March 29th, 2020, 08:35 Let's see, suppose we pick a convex function, say$f(x)=x^2$. It's convex isn't it? Does$\lim\limits_{x\to +\infty}f'(x)$exist? (Wondering) ... 21 replies | 404 view(s) • March 29th, 2020, 07:38 Hey mathmari!! Let's start with: It follows from$\lim\limits_{x\rightarrow +\infty}f(x)=\ell$that$\lim\limits_{x\to +\infty}f'(x)=0$... 21 replies | 404 view(s) • March 27th, 2020, 23:42 Here is this week's POTW: ----- Find the minimum value of$(u-v)^2+\left(\sqrt{2-u^2}-\dfrac{9}{v}\right)^2$for$0<u<\sqrt{2}$and$v>0\$. ...
1 replies | 128 view(s)
• March 27th, 2020, 23:38
Hi MHB! I have decided to extend the deadline by another week so that our members can give this problem another shot and I am looking forward to...
1 replies | 222 view(s)
• March 26th, 2020, 21:12
I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.4: Basis ... ......
0 replies | 137 view(s)
• March 26th, 2020, 20:42
Have you tried looking at \lim_{n \to \infty} \dfrac{a_{n + 1}}{a_n}? -Dan
2 replies | 74 view(s)
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#### March 14th, 2020

Ranks Showcase - 16 Ranks
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Name: MHB University Math Award (2018)
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