Differantiation proof question

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In summary, the conversation discusses a function f(x) that is differentiable on the interval (a,+infinity) and the given condition that lim [f(x)]/x = 0 as x->+infinity. The question is to prove that lim inf |f'(x)|=0 as x->+infinity. The solution involves using the mean value theorem and applying it to the interval [x,2x] as x approaches infinity. However, the person asking the question is unsure about using L'Hospital's rule in this proof.
  • #1
transgalactic
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there is a function f(x) which is differentiable on (a,+infinity)
suppose lim [f(x)]/x =0 as x->+infinity
prove that lim inf |f'(x)|=0 as x->+infinity ?

does this expression lim inf f'(x)=0 has to be true
if not
present a disproving example

?

i got this solution :
First consider [itex]\lim_{m\to\infty}\frac{f(2m)}{m}[/itex], let [itex]2m=x[/itex] and this limit becomes [itex]2\lim_{x\to\infty}\frac{f(x)}{x}=0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(2x)}{x}[/itex] exists and equals [itex]0[/itex]. So [itex]\lim_{x\to\infty}\frac{f(x)}{x}=0\implies \lim_{x\to\infty}\frac{f(2x)-f(x)}{x}=\lim_{x\to\infty}\frac{f(2x)-f(x)}{2x-x}=0[/itex]. So now consider the interval [itex][x,2x][/itex] and apply the MVT letting x approach infinity.

I tried to follow your logic.
and i got a few question regarding it??
http://img82.imageshack.us/my.php?image=14440292jz6.gif
 
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  • #2
L'Hospital rule?
 
  • #3
what about it?
 
  • #4
I thought you could use L'Hospital here... using that
you get
lim .. f'(x)/1 = 0
but f(x)/x may not be in the form of inf/inf etc.

I couldn't understand the solution.
 
Last edited:

1. What is the purpose of a differentiation proof?

A differentiation proof is a mathematical technique used to find the derivative of a function at a specific point. It is useful in many areas of science, particularly in physics and engineering, where it can be used to analyze the rate of change of a variable.

2. How do you write a differentiation proof?

To write a differentiation proof, you need to use the fundamental rules of differentiation, such as the power rule, product rule, and chain rule. You also need to understand the concept of limits and use them to determine the derivative at a specific point.

3. What is the difference between a differentiation proof and a derivative?

A differentiation proof is the process of finding the derivative of a function, while a derivative is the result of that process. Think of a differentiation proof as the steps you take to find the derivative, and the derivative as the final answer.

4. Can you give an example of a differentiation proof?

Sure, let's say we have the function f(x) = 2x^2 + 3x + 5. To find the derivative, we first apply the power rule to get f'(x) = 4x + 3. Then, we can use the derivative rules for addition and constant multiplication to get f'(x) = 4x + 3. This is an example of a differentiation proof.

5. When is a differentiation proof used in real life?

A differentiation proof is used in many real-life scenarios, such as calculating the speed of an object at a specific time or analyzing the growth rate of a population. It is also used in fields like economics and biology to understand the relationship between different variables.

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