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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:26:44
Please 🙏
1
0
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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:29:10
Hello?
1
0
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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:30:46
I want to make g have no indices
1
0
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507
help-10
e4fa1722f9124db4833f519852c0101b
293,744,814,472,822,800
Bunnies_
02/05/2022 08:31:14
you'll want to square both sides
0
1
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507
help-10
e4fa1722f9124db4833f519852c0101b
321,445,119,171,756,000
Shuri2060
02/05/2022 08:31:14
no
0
1
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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:31:44
oh yes
1
0
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help-10
e4fa1722f9124db4833f519852c0101b
321,445,119,171,756,000
Shuri2060
02/05/2022 08:32:29
$$\left(g ^\frac{1}{2}\right)^\frac{-1}{2}=g^\frac{-1}{4}$$
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help-10
e4fa1722f9124db4833f519852c0101b
510,789,298,321,096,700
TeXit
02/05/2022 08:32:51
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
e4fa1722f9124db4833f519852c0101b
321,445,119,171,756,000
Shuri2060
02/05/2022 08:32:57
You apply the same exponents to both sides
0
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help-10
e4fa1722f9124db4833f519852c0101b
321,445,119,171,756,000
Shuri2060
02/05/2022 08:33:10
so one of your first ones just doesnt work out cus of this
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help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:33:22
Oh ic
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0
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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:33:24
Thanks
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0
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507
help-10
e4fa1722f9124db4833f519852c0101b
886,259,475,911,180,300
uuuu
02/05/2022 08:33:28
.close
1
0
293,744,814,472,822,800
Bunnies_
1349
Bunnies_
false
508
help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:19:47
is the inverse of y=f(x) x=f(y)
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:20:50
no
0
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508
help-10
e76c0d60eed549b6884fca045b5cab44
521,326,998,195,273,700
shreksdad
02/05/2022 09:21:29
y = f(x) makes no sense, they serve similar purpose
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:21:38
hmm
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0
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:21:43
alright
1
0
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:22:26
wait
1
0
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help-10
e76c0d60eed549b6884fca045b5cab44
521,326,998,195,273,700
shreksdad
02/05/2022 09:22:27
f(x) means function
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help-10
e76c0d60eed549b6884fca045b5cab44
675,340,001,483,030,500
tam
02/05/2022 09:22:30
@rechonchota its x=g(y)
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:22:39
yes thats why i deleted it cause ik that
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help-10
e76c0d60eed549b6884fca045b5cab44
521,326,998,195,273,700
shreksdad
02/05/2022 09:22:44
Ah ok
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:22:47
srry i am learning about inverse functions
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:22:53
and to make an inverse function
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:22:56
the inverse of `f` if `f^(-1)` where `f(x) = y` and `(f^(-1))(y) = x`
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:23:17
you rearange to find x
1
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:23:22
then you change x to y
1
0
835,603,021,425,148,000
rechonchota
2320
rechonchota
false
508
help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:23:26
and `f^(-1)` is just a notation for another function (like `g(y)`)
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:23:31
ahh yes
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:23:47
why do we change all x to make y
1
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:24:03
like the inverse of y= 3/(x-1)
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508
help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:24:22
is x= (3+y)/y
1
0
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:24:28
but you chnage the y to x
1
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:25:20
`3/(x-1) = y` means `x = (y + 3)/y`
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:25:39
and `x = g(y) = (y+3)/y` is the inverse function
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:02
oh
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0
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:06
but you can change y to x
1
0
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:26:08
but since you usually use `x` as an argument and `y` as a function's value
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:10
and say it is f-1
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:26:11
you just rename it
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help-10
e76c0d60eed549b6884fca045b5cab44
586,955,478,529,998,800
rept1d
02/05/2022 09:26:17
so it is `y = g(x) = (x+3)/x`
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:28
perfect
1
0
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:34
idky im getting so confused
1
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:38
this has cleared it up
1
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help-10
e76c0d60eed549b6884fca045b5cab44
835,603,021,425,148,000
rechonchota
02/05/2022 09:26:41
thanks buddy
1
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509
help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:36:08
How valid is this proof? "Suppose there exists $N_0$ such that $s_n \le t_n \forall n > N_0$. Prove that if lim$t_n$ and lim$s_n$ exists, then lim$s_n$ $\le$ lim$t_n$." Proof: \\\\ Let lim$t_n$ = t and lim$s_n$ = s. Let $\epsilon > 0$, and observe that $|t_n - t| < \epsilon$ $\forall n > N_0$, $|s_n - s| < \epsilon$ $...
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509
help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 10:38:50
**dulg**
https://cdn.discordapp.c…244354711552.png
197712244354711552.png
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:39:57
\lim f(x)
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:40:00
looks a lot better in latex.
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:40:20
i will keep that in mind, thanks
1
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:41:15
https://cdn.discordapp.c…0640/unknown.png
unknown.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:41:24
How does this follow?
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:41:51
you basically add - s -t to both sides
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:41:58
oh nvm
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:42:02
i was reading -s+t in head
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:42:38
https://i.imgur.com/k7S46YM.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:42:49
How does this follow from the above?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:43:40
If s is greater than s_n or t greater than t_n
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:43:44
then ?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:45:47
=== Something helpful for this proof might be to draw a rough diagram
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 10:45:54
And to mark epsilon
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:57:07
Good point. I kind of assume it since the textbook already went through it. Because of how absolute value is defined, where $|a| = a$ if a > 0, and $|a| = -a$ if a < 0, it follows that $-|a| \le a \le |a|$, since either $a = |a|$ or $a = -|a|$. Thus: $-|s_n - s| \le s_n - s \le |s_n - s|$ $\Rightarrow$ $-|s_n - s| -t ...
1
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509
help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 10:57:12
**dulg**
https://cdn.discordapp.c…244354711552.png
197712244354711552.png
0
1
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509
help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:57:13
. @Shuri2060 ok wait
1
0
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:57:19
i am a bit stumped
1
0
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:57:33
realized something is a bit off halfway thru my last latex post
1
0
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 10:58:37
wait for me while i post this idea on how to prove $|s_n - s| - t \le |t_n - t| -s$
1
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509
help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 10:58:40
**dulg**
https://cdn.discordapp.c…244354711552.png
197712244354711552.png
0
1
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509
help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:04:30
Consider case 1, where $s_n - s -t = |s_n - s| - t$. $s_n \le t_n$ implies $|s_n - s| - t \le t_n - t - s \le |t_n - t| -s$. \\\\ Case 2, where I got stuck because I am not sure :(. Following the same process leads us to $s_n - s - t = -|s_n - s| - t \le t_n - t - s \le |t_n - t| - s$
1
0
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509
help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:04:33
**dulg**
https://cdn.discordapp.c…244354711552.png
197712244354711552.png
0
1
null
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509
help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:05:26
The ambiguity comes from where $|s_n - s| - t$ would fit in Case 2
1
0
321,445,119,171,756,000
Shuri2060
9525
a μ-er
false
509
help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:05:28
**dulg**
https://cdn.discordapp.c…244354711552.png
197712244354711552.png
0
1
null
null
null
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null
509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:06:33
I would say to restart your proof
0
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:06:43
Because there really is no need to split cases
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:06:49
What you need to do
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:06:52
is draw a rough diagram
0
1
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509
help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:07:03
and indicate what you want to show with epsilon pictorially
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:07:15
what would that look like?
1
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:07:33
I have never imagined real analysis using pictures
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:07:47
hmmm
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:08:38
the entire proof? or this absolute value thing?
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:09:08
https://cdn.discordapp.c…3748/unknown.png
unknown.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:09:19
Maybe all, cus there wasn't that much progress
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:09:37
I'm just saying restart in terms of ideas
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:09:42
Stuff will be reused
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:11:05
In particular, like you said
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:11:18
https://cdn.discordapp.c…2616/unknown.png
unknown.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:11:20
We will take this
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:11:39
Using this, we need to prove s <= t
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:12:01
idk if the picture will help you see what you need to do with these 2 inequalities
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:13:42
not sure about the picture
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help-10
209372b213bd479ca6cc76c57845c087
197,712,244,354,711,550
dulg
02/05/2022 11:14:05
i suppose they are converging to a set number, and one is def greater than the other
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:14:35
Try to draw your own
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:14:45
and mark an epsilon, basically
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:15:21
@dulg ah actually, I see an error in your inital proof
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:15:45
$$(\forall n > N_0)(\abs{s_n - s} < \varepsilon)$$ $$(\forall n > N_0)(\abs{t_n - t} < \varepsilon)$$
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help-10
209372b213bd479ca6cc76c57845c087
510,789,298,321,096,700
TeXit
02/05/2022 11:15:48
**Shuri2060**
https://cdn.discordapp.c…119171756033.png
321445119171756033.png
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:15:54
You make these claims.
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help-10
209372b213bd479ca6cc76c57845c087
321,445,119,171,756,000
Shuri2060
02/05/2022 11:16:01
They are not necessarily true --- think carefully why.
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