SUMMARY
If the derivative f'(x) is greater than 0 for all real values of x, it does not guarantee that the function f(x) increases without bounds. The function f(x) = 2x/sqrt(x² + 2) serves as a counterexample, where its derivative f'(x) = 4/(x² + 2)^(3/2) is always positive, yet f(x) approaches a horizontal asymptote of 2 as x approaches infinity. This demonstrates that a positive derivative does not imply unbounded growth.
PREREQUISITES
- Understanding of calculus concepts, particularly derivatives
- Familiarity with limits and asymptotic behavior
- Knowledge of horizontal asymptotes
- Ability to differentiate functions
NEXT STEPS
- Study the concept of horizontal asymptotes in calculus
- Learn about the implications of derivatives on function behavior
- Explore examples of functions with positive derivatives that do not increase without bounds
- Investigate the relationship between limits and derivatives
USEFUL FOR
Students of calculus, mathematics educators, and anyone interested in understanding the nuances of function behavior in relation to derivatives.