Lim f'(x)=0 implies lim f(x)/x = 0

  • Thread starter Kreutzfelt
  • Start date
  • #1

Homework Statement


Given that the function [itex]f[/itex] is differentiable on the interval [itex](a,\infty)[/itex] and that [itex]\lim_{x\rightarrow\infty}f'(x)= 0[/itex]. Show that [itex]\lim_{x\rightarrow\infty}\frac{f(x)}{x}= 0[/itex].

Homework Equations





The Attempt at a Solution


I have a pretty good intuition of why this is true: As f(x) approaches infinity, its derivative approaches 0. This means that, at some point, the function slope becomes less than that of x.
This means that when x approaches infinity, x is dominant over f(x) and thus f(x)/x => 0.

I can't come up with any strict mathematical proof though, so I would be very greatful if someone could help me with this.
 

Answers and Replies

  • #2
34,158
5,778

Homework Statement


Given that the function [itex]f[/itex] is differentiable on the interval [itex](a,\infty)[/itex] and that [itex]\lim_{x\rightarrow\infty}f'(x)= 0[/itex]. Show that [itex]\lim_{x\rightarrow\infty}\frac{f(x)}{x}= 0[/itex].

Homework Equations





The Attempt at a Solution


I have a pretty good intuition of why this is true: As f(x) approaches infinity, its derivative approaches 0. This means that, at some point, the function slope becomes less than that of x.
This means that when x approaches infinity, x is dominant over f(x) and thus f(x)/x => 0.

I can't come up with any strict mathematical proof though, so I would be very greatful if someone could help me with this.
L' Hopital's Rule?
 
  • #3
L'Hopital's rule is unfortunately not allowed in this particular problem.
 

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