$lim_x to 2 fraccosleft(fracpixright)x-2$ without using De L'Hospital [duplicate]
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This question already has an answer here:
Limit as $xto 2$ of $fraccos(frac pi x)x-2 $
5 answers
$$lim_x to 2 fraccosleft(fracpixright)x-2$$
This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or Squeeze Theorem. For each attempt, I expected at least one of the well-known limits $lim_x to 0 fracsinxx$ or $lim_x to 0 fraccosx - 1x$ to come into use. They frequently did in my attempts but did not go anywhere. I have also been experimented substitutions but in vain.
How can this be solved without using L'Hospital's Rule?
limits-without-lhopital
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
marked as duplicate by Jyrki Lahtonen, Nosrati, Lord Shark the Unknown, user91500, Jendrik Stelzner Sep 7 at 10:31
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |Â
up vote
1
down vote
favorite
This question already has an answer here:
Limit as $xto 2$ of $fraccos(frac pi x)x-2 $
5 answers
$$lim_x to 2 fraccosleft(fracpixright)x-2$$
This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or Squeeze Theorem. For each attempt, I expected at least one of the well-known limits $lim_x to 0 fracsinxx$ or $lim_x to 0 fraccosx - 1x$ to come into use. They frequently did in my attempts but did not go anywhere. I have also been experimented substitutions but in vain.
How can this be solved without using L'Hospital's Rule?
limits-without-lhopital
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
marked as duplicate by Jyrki Lahtonen, Nosrati, Lord Shark the Unknown, user91500, Jendrik Stelzner Sep 7 at 10:31
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This question already has an answer here:
Limit as $xto 2$ of $fraccos(frac pi x)x-2 $
5 answers
$$lim_x to 2 fraccosleft(fracpixright)x-2$$
This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or Squeeze Theorem. For each attempt, I expected at least one of the well-known limits $lim_x to 0 fracsinxx$ or $lim_x to 0 fraccosx - 1x$ to come into use. They frequently did in my attempts but did not go anywhere. I have also been experimented substitutions but in vain.
How can this be solved without using L'Hospital's Rule?
limits-without-lhopital
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
This question already has an answer here:
Limit as $xto 2$ of $fraccos(frac pi x)x-2 $
5 answers
$$lim_x to 2 fraccosleft(fracpixright)x-2$$
This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or Squeeze Theorem. For each attempt, I expected at least one of the well-known limits $lim_x to 0 fracsinxx$ or $lim_x to 0 fraccosx - 1x$ to come into use. They frequently did in my attempts but did not go anywhere. I have also been experimented substitutions but in vain.
How can this be solved without using L'Hospital's Rule?
This question already has an answer here:
Limit as $xto 2$ of $fraccos(frac pi x)x-2 $
5 answers
limits-without-lhopital
limits-without-lhopital
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
edited Sep 7 at 10:30
Jendrik Stelzner
7,69121137
7,69121137
asked Sep 6 at 18:22
hatinacat2000
294
asked Sep 6 at 18:22
hatinacat2000
294
asked Sep 6 at 18:22
hatinacat2000
294
294
marked as duplicate by Jyrki Lahtonen, Nosrati, Lord Shark the Unknown, user91500, Jendrik Stelzner Sep 7 at 10:31
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Jyrki Lahtonen, Nosrati, Lord Shark the Unknown, user91500, Jendrik Stelzner Sep 7 at 10:31
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04
add a comment |Â
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
7
down vote
By definition of derivative, if $f(x)=cosleft(fracpi xright)$, then$$lim_xto2fraccosleft(fracpi xright)x-2=f'(2)=fracpi4.$$
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
3
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
add a comment |Â
up vote
1
down vote
With $dfracpi x=dfracpi2-t$,
$$limlimits_x to 2 fraccosleft(dfracpixright)x-2=limlimits_t to 0fracpi-2t4fracsinleft(tright)t.$$
No need to say more.
answered Sep 6 at 18:36
Yves Daoust
115k666209
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
add a comment |Â
up vote
1
down vote
This expression corresponds with the definition of the derivative. Recall that $$f'(x) = limlimits_yto xfracf(y)-f(x)y-x$$ Using the fact that $cosleft(fracpi2right) = 0$, this expression turns out to be the derivative of $cos(fracpix)$ at $x = 2$.
EDIT: Sorry, used the wrong function.
edited Sep 6 at 18:54
Bernard
112k635104
answered Sep 6 at 18:25
Josh
1789
add a comment |Â
up vote
1
down vote
As an alternative, without derivatives, we have that
$$ fraccosleft(fracpixright)x-2
= fracfracpi2-fracpixx-2fracsinleft(fracpi2-fracpixright)fracpi2-fracpix
=fracpi(x-2)2x(x-2)fracsinleft(fracpi2-fracpixright)fracpi2-fracpix=$$
$$=fracpi2xfracsinleft(fracpi2-fracpixright)fracpi2-fracpixto fracpi4cdot 1 =fracpi4$$
answered Sep 6 at 19:55
gimusi
73.9k73889
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
By definition of derivative, if $f(x)=cosleft(fracpi xright)$, then$$lim_xto2fraccosleft(fracpi xright)x-2=f'(2)=fracpi4.$$
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
3
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
add a comment |Â
up vote
7
down vote
By definition of derivative, if $f(x)=cosleft(fracpi xright)$, then$$lim_xto2fraccosleft(fracpi xright)x-2=f'(2)=fracpi4.$$
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
3
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
add a comment |Â
up vote
7
down vote
up vote
7
down vote
By definition of derivative, if $f(x)=cosleft(fracpi xright)$, then$$lim_xto2fraccosleft(fracpi xright)x-2=f'(2)=fracpi4.$$
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
By definition of derivative, if $f(x)=cosleft(fracpi xright)$, then$$lim_xto2fraccosleft(fracpi xright)x-2=f'(2)=fracpi4.$$
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
edited Sep 6 at 18:32
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
answered Sep 6 at 18:26
José Carlos Santos
123k17101186
123k17101186
3
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
add a comment |Â
3
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
3
3
intuition is a gift+
– mrs
Sep 6 at 18:28
intuition is a gift+
– mrs
Sep 6 at 18:28
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
Answer should be $pi/4$.
– StammeringMathematician
Sep 6 at 18:31
1
1
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
@StammeringMathematician I've edited my answer. Thank you.
– José Carlos Santos
Sep 6 at 18:32
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
IMO, using this trick is a little cheating. There is very little difference between $(f(x)-f(x_0))/(x-x_0)to f'(x_0)$ and $(f(x)-f(x_0))/(x-x_0)to f'(x_0)/1$.
– Yves Daoust
Sep 6 at 19:05
add a comment |Â
up vote
1
down vote
With $dfracpi x=dfracpi2-t$,
$$limlimits_x to 2 fraccosleft(dfracpixright)x-2=limlimits_t to 0fracpi-2t4fracsinleft(tright)t.$$
No need to say more.
answered Sep 6 at 18:36
Yves Daoust
115k666209
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
add a comment |Â
up vote
1
down vote
With $dfracpi x=dfracpi2-t$,
$$limlimits_x to 2 fraccosleft(dfracpixright)x-2=limlimits_t to 0fracpi-2t4fracsinleft(tright)t.$$
No need to say more.
answered Sep 6 at 18:36
Yves Daoust
115k666209
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
add a comment |Â
up vote
1
down vote
up vote
1
down vote
With $dfracpi x=dfracpi2-t$,
$$limlimits_x to 2 fraccosleft(dfracpixright)x-2=limlimits_t to 0fracpi-2t4fracsinleft(tright)t.$$
No need to say more.
answered Sep 6 at 18:36
Yves Daoust
115k666209
With $dfracpi x=dfracpi2-t$,
$$limlimits_x to 2 fraccosleft(dfracpixright)x-2=limlimits_t to 0fracpi-2t4fracsinleft(tright)t.$$
No need to say more.
answered Sep 6 at 18:36
Yves Daoust
115k666209
answered Sep 6 at 18:36
Yves Daoust
115k666209
answered Sep 6 at 18:36
Yves Daoust
115k666209
answered Sep 6 at 18:36
Yves Daoust
115k666209
115k666209
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
add a comment |Â
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
Can you please show your work? I have tried your substitution but am not able to get this result. Also, how did you come up with this substitution?
– hatinacat2000
Sep 7 at 5:03
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
@hatinacat2000: this is elementary algebra. And seeing the shape of the expression ($0/0$ with a trigonometric function), I understood that it could be turned to $sin t/t$ by an appropriate transform.
– Yves Daoust
Sep 7 at 7:12
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
I can get $frac sin tt $, but not $(pi - 2t)/4$
– hatinacat2000
Sep 7 at 17:19
1
1
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
Edit: success after more substitutions
– hatinacat2000
Sep 7 at 17:50
add a comment |Â
up vote
1
down vote
This expression corresponds with the definition of the derivative. Recall that $$f'(x) = limlimits_yto xfracf(y)-f(x)y-x$$ Using the fact that $cosleft(fracpi2right) = 0$, this expression turns out to be the derivative of $cos(fracpix)$ at $x = 2$.
EDIT: Sorry, used the wrong function.
edited Sep 6 at 18:54
Bernard
112k635104
answered Sep 6 at 18:25
Josh
1789
add a comment |Â
up vote
1
down vote
This expression corresponds with the definition of the derivative. Recall that $$f'(x) = limlimits_yto xfracf(y)-f(x)y-x$$ Using the fact that $cosleft(fracpi2right) = 0$, this expression turns out to be the derivative of $cos(fracpix)$ at $x = 2$.
EDIT: Sorry, used the wrong function.
edited Sep 6 at 18:54
Bernard
112k635104
answered Sep 6 at 18:25
Josh
1789
add a comment |Â
up vote
1
down vote
up vote
1
down vote
This expression corresponds with the definition of the derivative. Recall that $$f'(x) = limlimits_yto xfracf(y)-f(x)y-x$$ Using the fact that $cosleft(fracpi2right) = 0$, this expression turns out to be the derivative of $cos(fracpix)$ at $x = 2$.
EDIT: Sorry, used the wrong function.
edited Sep 6 at 18:54
Bernard
112k635104
answered Sep 6 at 18:25
Josh
1789
This expression corresponds with the definition of the derivative. Recall that $$f'(x) = limlimits_yto xfracf(y)-f(x)y-x$$ Using the fact that $cosleft(fracpi2right) = 0$, this expression turns out to be the derivative of $cos(fracpix)$ at $x = 2$.
EDIT: Sorry, used the wrong function.
edited Sep 6 at 18:54
Bernard
112k635104
answered Sep 6 at 18:25
Josh
1789
edited Sep 6 at 18:54
Bernard
112k635104
edited Sep 6 at 18:54
Bernard
112k635104
edited Sep 6 at 18:54
Bernard
112k635104
112k635104
answered Sep 6 at 18:25
Josh
1789
answered Sep 6 at 18:25
Josh
1789
answered Sep 6 at 18:25
Josh
1789
1789
add a comment |Â
add a comment |Â
up vote
1
down vote
As an alternative, without derivatives, we have that
$$ fraccosleft(fracpixright)x-2
= fracfracpi2-fracpixx-2fracsinleft(fracpi2-fracpixright)fracpi2-fracpix
=fracpi(x-2)2x(x-2)fracsinleft(fracpi2-fracpixright)fracpi2-fracpix=$$
$$=fracpi2xfracsinleft(fracpi2-fracpixright)fracpi2-fracpixto fracpi4cdot 1 =fracpi4$$
answered Sep 6 at 19:55
gimusi
73.9k73889
add a comment |Â
up vote
1
down vote
As an alternative, without derivatives, we have that
$$ fraccosleft(fracpixright)x-2
= fracfracpi2-fracpixx-2fracsinleft(fracpi2-fracpixright)fracpi2-fracpix
=fracpi(x-2)2x(x-2)fracsinleft(fracpi2-fracpixright)fracpi2-fracpix=$$
$$=fracpi2xfracsinleft(fracpi2-fracpixright)fracpi2-fracpixto fracpi4cdot 1 =fracpi4$$
answered Sep 6 at 19:55
gimusi
73.9k73889
add a comment |Â
up vote
1
down vote
up vote
1
down vote
As an alternative, without derivatives, we have that
$$ fraccosleft(fracpixright)x-2
= fracfracpi2-fracpixx-2fracsinleft(fracpi2-fracpixright)fracpi2-fracpix
=fracpi(x-2)2x(x-2)fracsinleft(fracpi2-fracpixright)fracpi2-fracpix=$$
$$=fracpi2xfracsinleft(fracpi2-fracpixright)fracpi2-fracpixto fracpi4cdot 1 =fracpi4$$
answered Sep 6 at 19:55
gimusi
73.9k73889
As an alternative, without derivatives, we have that
$$ fraccosleft(fracpixright)x-2
= fracfracpi2-fracpixx-2fracsinleft(fracpi2-fracpixright)fracpi2-fracpix
=fracpi(x-2)2x(x-2)fracsinleft(fracpi2-fracpixright)fracpi2-fracpix=$$
$$=fracpi2xfracsinleft(fracpi2-fracpixright)fracpi2-fracpixto fracpi4cdot 1 =fracpi4$$
answered Sep 6 at 19:55
gimusi
73.9k73889
edited Sep 7 at 0:06
answered Sep 6 at 19:55
gimusi
73.9k73889
answered Sep 6 at 19:55
gimusi
73.9k73889
answered Sep 6 at 19:55
gimusi
73.9k73889
73.9k73889
add a comment |Â
add a comment |Â
Someone did downvote all answers, but s/he forgot to do downvote the question $;)$.
– Nosrati
Sep 7 at 5:15
@Nosrati: I am OP and I did not mark a single answer down, so please be more hesitant to pass judgement
– hatinacat2000
Sep 7 at 17:20
I didn't say you are downwoter! I said that for fun dear. Anyway I did delete my answer. $:)$ @hanti
– Nosrati
Sep 7 at 20:04