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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 | 82 view(s) • Today, 09:14 It's like this: $$\Big(1-\frac 12 x^2 + \ldots\Big)\Big(1+\frac 12 (2x)^2 - \ldots\Big)\\ =1\cdot 1 -\frac 12 x^2 \cdot 1+1\cdot \frac 12 (2x)^2 -... 7 replies | 82 view(s) • Today, 07:04 The series expansion of \cos x = 1 - \frac 12x^2 + \ldots. And the expansion of \frac 1{1-x} = 1+x+x^2+\ldots So:$$\frac{\cos(x)}{\cos(2x)}... 7 replies | 82 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 | 162 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, 04:35 mathmari replied to a thread Limits & properties in Analysis Ok!! As for the second question, why is the limit always zero? (Wondering) 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, 03:57 mathmari replied to a thread Limits & properties in Analysis OK, so it's graph is a decreasing function, right? Or can we say something more specifically? (Wondering) 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, 17:21 mathmari replied to a thread Limits & properties in Analysis I don't see how we get the strict inequality. Is maybe the result wrong and it should be monotone instead of strictly monotone? (Wondering) 21 replies | 404 view(s) • March 29th, 2020, 14:38 Indeed. (Thinking) 21 replies | 404 view(s) • March 29th, 2020, 14:31 mathmari replied to a thread Limits & properties in Analysis Ok! That means that$f$is monotone and escpecially descreasing, right? To get that$f$is strictly monotone, do we have to get$f'(x)<0$instead of... 21 replies | 404 view(s) • March 29th, 2020, 14:26 Yep. (Nod) 21 replies | 404 view(s) • March 29th, 2020, 14:18 mathmari replied to a thread Limits & properties in Analysis Yes, because then from$\ell\geq f(y)+f'(y)(+\infty-y)$we get$-\infty$at the right side of the inequality, and so$\ell$can be real. ... 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, 13:52 mathmari replied to a thread Limits & properties in Analysis Ah$(-\infty)\cdot (+\infty)$is also an undefined form, isn't it? So$f'(y)$is either$0$or$-\infty$, right? (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, 13:14 mathmari replied to a thread Limits & properties in Analysis So that we get that$\ell$is not infinity we have to get an undefined form, one such form is when infinity is multiplied with zero, so$f'(y)$must... 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$\pm\inftyeither. ... 21 replies | 404 view(s) • March 29th, 2020, 12:28 mathmari replied to a thread Limits & properties in Analysis So we have the following: \begin{align*}f(x)\geq f(y)+f'(y)(x-y)&\Rightarrow \lim_{x\rightarrow +\infty}f(x)\geq \lim_{x\rightarrow... 21 replies | 404 view(s) • March 29th, 2020, 10:27 Ah okay. But that is not the case now is it? (Wondering) Sounds like a plan. (Nod) 21 replies | 404 view(s) • March 29th, 2020, 09:06 mathmari replied to a thread Limits & properties in Analysis At the previous exerciseL$belong to$\mathbb{R}\cup \{\pm \infty\}$. So since$f'(x)=2x$and the limit is equal to$+\infty$and so it exists.... 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, 08:09 mathmari replied to a thread Limits & properties in Analysis In a previous exercise I showed that $$\lim_{x\rightarrow +\infty}(f(x+1)-f(x))=L \Rightarrow \lim_{x\rightarrow +\infty}f'(x)=L$$ if we know that... 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 29th, 2020, 05:52 mathmari started a thread Limits & properties in Analysis Hey!! :o Could you give me a hint how to prove the following statements? (Wondering) Let$f:\mathbb{R}\rightarrow \mathbb{R}\$ be...
21 replies | 404 view(s)
• March 28th, 2020, 16:11
Welcome to the forum! I guess that "between" is supposed to mean "among", but I am not sure about "even". If it means "at least", then the...
2 replies | 232 view(s)
• March 26th, 2020, 14:09
Nice... (Smirk) Thank you very much!!! (Blush)
12 replies | 382 view(s)
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