• 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 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 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, 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 25th, 2020, 17:42 Yep. (Nod) I'd say 'and this is confirmed...' rather than 'since this is confimed...' though. (Emo) 12 replies | 381 view(s) • March 25th, 2020, 15:50 Ah good. (Whew) Does the solution for$x$match the given general solution? (Wondering) 12 replies | 381 view(s) • March 25th, 2020, 15:40 How did you get$x_{1}=\frac{1}{3}$? (Wondering) 12 replies | 381 view(s) • March 25th, 2020, 15:06 Don't we have$x_2=h(y)$so that$f(x_2)=f(h(y))=y$? And therefore$g(f(x_2))=g(y)=x_1$? (Wondering) 4 replies | 145 view(s) • March 25th, 2020, 14:35 Suppose it is correct, then what are the solutions of$Ax=\begin{bmatrix}1\\3\end{bmatrix}$? (Thinking) 12 replies | 381 view(s) • March 25th, 2020, 14:33 Hey mathmari!! How about a proof by contradiction? That is, suppose$g\not\equiv h$, then there must be an$y\in B$such that$g(y)\ne h(y)$,... 4 replies | 145 view(s) More Activity ### 7 Visitor Messages 1. Welcome back to MHB, friend! I'm so glad to see your return to our forum again!  2. Sorry... There seems to be an echo in here [my fault!   ] 3. Hello K! I just wanted to say, i think you do an absolutely sterling job posting all sorts of brain-teasers on the Puzzles board. Keep up the great work! Gethin 4. Hello K! I just wanted to say, i think you do an absolutely sterling job posting all sorts of brain-teasers on the Puzzles board. Keep up the great work! Gethin 5. Hi there, I didn't realize Mark has already taught you how to type square root in LaTeX... and what I mentioned in the challenge thread that we both participated is essentially the same as Mark's ... 6. Hey Kali, To create a square root using$\LaTeX$, use the command \sqrt{} where the radicand goes in the braces. For example: \sqrt{x^2+y^2} gives you$\sqrt{x^2+y^2}$To create other roots, use the command \sqrt[]{} where the degree of the roots goes in the square brackets. For example: \sqrt[n]{a^m}=a^{\frac{m}{n}} gives you$\sqrt[n]{a^m}=a^{\frac{m}{n}}$Hope this helps, and please do not hesitate to ask me if you need to know any other$\LaTeX\$ commands. Best Regards,

Mark.
7. Hello and welcome to MHB, kaliprasad! If you have any questions or comments, pleases feel free to address them to me or another staff member. We are happy to help, and we look forward to your participation here!

Best Regards,

Mark.
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Ranks Showcase - 4 Ranks
Icon Image Description  Name: Secondary School/High School POTW Award (2016)
 Issue time: January 9th, 2017 14:43 Issue reason:  Name: MHB Challenges Solver Award (2016)
 Issue time: January 9th, 2017 14:42 Issue reason:  Name: Secondary School/High School POTW Award (2015)
 Issue time: January 7th, 2016 11:12 Issue reason:  Name: MHB Challenges Solver Award (2015)
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