• Yesterday, 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 | 79 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 | 79 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 | 280 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 | 280 view(s)
• March 30th, 2020, 04:26
Yep. (Nod) And no, nothing more specific.
21 replies | 280 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 | 280 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 | 280 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 | 280 view(s)
• March 29th, 2020, 14:38
Indeed. (Thinking)
21 replies | 280 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 | 280 view(s)
• March 29th, 2020, 14:26
Yep. (Nod)
21 replies | 280 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 | 280 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 | 280 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 | 280 view(s)
• March 29th, 2020, 13:19
That is a possibility yes. What happens if $f'(y)$ is negative? (Wondering)
21 replies | 280 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 | 280 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 | 280 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 | 280 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 | 280 view(s)
• March 29th, 2020, 09:06
mathmari replied to a thread Limits & properties in Analysis
At the previous exercise $L$ 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 | 280 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 | 280 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 | 280 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 | 280 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 | 280 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 $v>0$. ...
0 replies | 80 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 | 200 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 | 99 view(s)
• March 26th, 2020, 14:09
Nice... (Smirk) Thank you very much!!! (Blush)
12 replies | 318 view(s)
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