- #1

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[tex] y(\infty) =0 [/tex] and [tex] y'(\infty) =1 [/tex] ?

the inverse case, a function that tends to 1 for big x and whose derivative tends to 0 is quite obvious but this case i am not sure if there will exist

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- Thread starter zetafunction
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- #1

- 391

- 0

[tex] y(\infty) =0 [/tex] and [tex] y'(\infty) =1 [/tex] ?

the inverse case, a function that tends to 1 for big x and whose derivative tends to 0 is quite obvious but this case i am not sure if there will exist

- #2

CompuChip

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[tex]\lim_{x \to \infty} y(x) = 0[/tex] and [tex]\lim_{x \to \infty} y'(x) = 1[/tex] ?

Actually, I think that for a smooth function to have a limit at infinity, the derivative should have limit 0 (at least that's what my intuition tells me: for the function to have a limit at infinity, it should become progressively more flat, so it doesn't run away from its limit value).

I have some other work now, but I will try to prove that rigorously later (if you want, give it a try yourself).

- #3

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It follows easily from MVT.

- #4

CompuChip

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After finishing my Saturday's to-do list I will take a completely blank paper of normal size and try again :)

- #5

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On the second thought, derivative doesn't have to go to zero, unfortunetely. Consider [tex]\frac{\sin (x^2)}{x}[/tex]. It clearly tends to zero, yet the derivative oscilates. Still, the derivative cannot tend to a nonzero number, and **this** follows from MVT for sure

Sorry for the mistake.

Edit: derivative of [tex]\frac{\sin (x^2)}{\sqrt{x}}[/tex] oscilates unboundedly, while the function goes to zero.

Sorry for the mistake.

Edit: derivative of [tex]\frac{\sin (x^2)}{\sqrt{x}}[/tex] oscilates unboundedly, while the function goes to zero.

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