Limit Theorem: Does f'(x) Approach 0 as x Goes to Infinity?

[vincent_vega]

suppose there is a function f(x), and it's limit as x goes to infinity is 0.

Is there a theorem that says it's derivative, f'(x), also approaches 0 as x goes to infinity?

Thanks.

 

[micromass]

Probably not, since it is not true. Consider

$$f(x)=\frac{\sin(x^2)}{x}$$

What is true, that if $f^\prime(x)$ has a limit, then this limit must be 0.

 

[Bacle2]

If f is differentiable in (-oo,oo) , use the mean value theorem:

f(b)-f(a)=f'(c)(b-a) . Maybe you can partition [0,oo)into [0,1],[0,2],...,[n,n+1].

Then:

f(1)=f(0)+f'(co) ;f(2)=f(1)+f'(c1) ;...f(n)=f(n-1)+f'(c_(n-1)).

Then you can find a closed form for f(n).

 

[haruspex]

If f is differentiable in (-oo,oo) , use the mean value theorem:

micromass's example shows this isn't going to work.

 

[Bacle2]

Of course I'm assuming f'(x) is defined as x-->oo , that is implied in my argument.

Basically, take [0,b] . Then

f(b)-f(0)=f'(c)(b)

If f' is defined everywhere and we let b-->oo , then the limit cannot be 0 unless f'(c) decreases to zero.

If the OP says "its derivative approaches 0 as x --> infinity" seems to me to assume that the derivative is defined as x-->oo.

 

[haruspex]

Of course I'm assuming f'(x) is defined as x-->oo , that is implied in my argument.

You mean, that the limit is defined? If your argument relies on that then it needs to be stated.

 

[Bacle2]

From the OP:" it's derivative, f'(x), also approaches 0 as x goes to infinity? "


This looks to me like an assumption that f'(x) is defined as x-->oo

 

[Bacle2]

Maybe the OP can clarify the conditions of the problem to eliminate ambiguity?

 

[haruspex]

From the OP:" its derivative, f'(x), also approaches 0 as x goes to infinity? "
This looks to me like an assumption that f'(x) is defined as x-->oo

No, that's what he's trying to prove. He didn't say "if f' approaches a limit, that limit is 0", so the most reasonable interpretation is that he wants to prove "f' approaches a limit, and that limit is 0".

 

[Bacle2]

It's not clear to me either way. f'(x) is said to exist without any qualification; I see

no reason to assume it exists in a specific subset of the real line only, nor reason

to assume otherwise. In your interpretation, why didn't the OP say something like

is f'(x) defined, and if so , what is its limit. He refers to f'(x) which states that f'(x)

exists. It may exist somewhere or everywhere.

The problem is posed sloppily ; I think out of basic manners, the OP should clarify.

 

[pwsnafu]

f'(x) is said to exist without any qualification

Then it does not follow that the limit $\lim_{x\rightarrow\infty}f'(x)$ is defined. The statement "f' exists" is limited to $x \in \mathbb{R}$ because the domain of f is the real numbers and not the extended reals. What happens as $x\rightarrow\infty$ is considered a separate condition, and must explicitly be mentioned.