bounded differentiable functions
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I need some help with the following problem:
Suppose $fcolon(0,infty) to mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) to 0$ as $x to +infty$, does this imply that $lim_{x to +infty}$ exists? Prove or give a counterexample.
It looks like it is false, but I am struggling to find the bounded counterexample.
real-analysis analysis derivatives examples-counterexamples
asked Dec 2 '18 at 17:02
PeterPeter
934
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add a comment |
$begingroup$
I need some help with the following problem:
Suppose $fcolon(0,infty) to mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) to 0$ as $x to +infty$, does this imply that $lim_{x to +infty}$ exists? Prove or give a counterexample.
It looks like it is false, but I am struggling to find the bounded counterexample.
real-analysis analysis derivatives examples-counterexamples
asked Dec 2 '18 at 17:02
PeterPeter
934
$endgroup$
add a comment |
$begingroup$
I need some help with the following problem:
Suppose $fcolon(0,infty) to mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) to 0$ as $x to +infty$, does this imply that $lim_{x to +infty}$ exists? Prove or give a counterexample.
It looks like it is false, but I am struggling to find the bounded counterexample.
real-analysis analysis derivatives examples-counterexamples
asked Dec 2 '18 at 17:02
PeterPeter
934
$endgroup$
I need some help with the following problem:
Suppose $fcolon(0,infty) to mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) to 0$ as $x to +infty$, does this imply that $lim_{x to +infty}$ exists? Prove or give a counterexample.
It looks like it is false, but I am struggling to find the bounded counterexample.
real-analysis analysis derivatives examples-counterexamples
real-analysis analysis derivatives examples-counterexamples
asked Dec 2 '18 at 17:02
PeterPeter
934
asked Dec 2 '18 at 17:02
PeterPeter
934
asked Dec 2 '18 at 17:02
PeterPeter
934
asked Dec 2 '18 at 17:02
PeterPeter
934
asked Dec 2 '18 at 17:02
PeterPeter
934
934
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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You can consider a function like $sin(sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
$endgroup$
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
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@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
You can consider a function like $sin(sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
$endgroup$
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
add a comment |
$begingroup$
You can consider a function like $sin(sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
$endgroup$
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
add a comment |
$begingroup$
You can consider a function like $sin(sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
$endgroup$
You can consider a function like $sin(sqrt{x+1})$; the derivative of this tends to $0$ but the function clearly oscillates.
It may be helpful for me to explain how I came up with this example. We want a bounded function that doesn't have a limit at infinity; the trig functions are my go to example. The issue is that the derivatives also don't have a limit. How do you make a derivative smaller? Stretch out the graph horizontally. How do you make the derivatives tend to zero? Make the stretch get worse and worse as you head towards infinity.
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
answered Dec 2 '18 at 17:04
BlarglFlargBlarglFlarg
2234
2234
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
add a comment |
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
Is there a reason for the +1 or sin(sqrt(x)) also work?
$endgroup$
– Peter
Dec 2 '18 at 17:14
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
$endgroup$
– BlarglFlarg
Dec 2 '18 at 17:15
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
$begingroup$
haha thanks @BlargFlarg!
$endgroup$
– Peter
Dec 2 '18 at 17:16
add a comment |
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