Another limit homework problem

Click For Summary
SUMMARY

The discussion centers on the mathematical limit question: Does lim_{x\rightarrow \infty} f(x)=0 imply lim_{x\rightarrow \infty} f'(x)=0? The user seeks clarification and proof regarding this relationship. A counterexample provided is f(x) = sin(x^2)/x, which demonstrates that the limit of the derivative does not necessarily approach zero even if the limit of the function itself does.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with derivatives and their properties
  • Knowledge of trigonometric functions and their behavior
  • Basic concepts of asymptotic analysis
NEXT STEPS
  • Study the properties of limits and derivatives in calculus
  • Research counterexamples in limit theory, particularly involving oscillatory functions
  • Explore the implications of L'Hôpital's Rule in limit evaluation
  • Examine advanced topics in asymptotic analysis and their applications
USEFUL FOR

Students of calculus, mathematicians, and educators seeking to deepen their understanding of limits and derivatives, particularly in the context of asymptotic behavior and counterexamples.

calculus_jy
Messages
56
Reaction score
0
This questions has been bothering me for quiet a while

Does lim_{x\rightarrow \infty} f(x)=0 \Rightarrow lim_{x\rightarrow \infty} f'(x)=0 if so can you please post a proof because its giving me a headache. If not would it please be possible for you to give me a counter example.

Many thanks
 
Last edited:
Physics news on Phys.org


calculus_jy said:
This questions has been bothering me for quiet a while

Does lim_{x\rightarrow \infty} f(x)=0 \Rightarrow lim_{x\rightarrow \infty} f'(x)=0 if so can you please post a proof because its giving me a headache. If not would it please be possible for you to give me a counter example.

Many thanks

\frac{sin(x^2)}{x}
 

Similar threads

Undergrad Calculate limits as distributions
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Some questions on l'Hôpital rules
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Limit of the product of these two functions
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Can Functions Have Complex Poles Beyond Infinitesimal Limits?
  • · Replies 10 ·
Replies
10
Views
2K
MHB Is the Convergence of These Sequences Correctly Determined?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Curiosity: there exists the exponential integral?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Limit of a trigonometric function
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Looking for Guarantees that the method of fixed-point iteration will work
  • · Replies 22 ·
Replies
22
Views
3K
Undergrad Proving that convexity implies second order derivative being positive
  • · Replies 6 ·
Replies
6
Views
2K