bounded differentiable functions
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 at 17:02
Peter
864
add a comment |
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 at 17:02
Peter
864
add a comment |
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 at 17:02
Peter
864
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 at 17:02
Peter
864
asked Dec 2 at 17:02
Peter
864
asked Dec 2 at 17:02
Peter
864
asked Dec 2 at 17:02
Peter
864
asked Dec 2 at 17:02
Peter
864
864
add a comment |
add a comment |
1 Answer
1
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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 at 17:04
BlarglFlarg
2234
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 at 17:04
BlarglFlarg
2234
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
add a comment |
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 at 17:04
BlarglFlarg
2234
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
add a comment |
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 at 17:04
BlarglFlarg
2234
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 at 17:04
BlarglFlarg
2234
answered Dec 2 at 17:04
BlarglFlarg
2234
answered Dec 2 at 17:04
BlarglFlarg
2234
answered Dec 2 at 17:04
BlarglFlarg
2234
2234
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
add a comment |
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
Is there a reason for the +1 or sin(sqrt(x)) also work?
– Peter
Dec 2 at 17:14
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
@Peter No reason in particular, I just don't like singularities anywhere near me if I can avoid it.
– BlarglFlarg
Dec 2 at 17:15
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
haha thanks @BlargFlarg!
– Peter
Dec 2 at 17:16
add a comment |
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