SUMMARY
The limit of the circumference as the power approaches infinity is defined by the equation $\displaystyle\lim_{p\to +\infty}C_p ((0,0); 1)=C_{\infty}((0,0);1)$. The discussion confirms that as $p$ increases, the expression $\displaystyle\lim_{p\to +\infty} \left|x \right|^p+\left|y \right|^p$ converges to $\max\{\left|x \right|, \left|y \right|\}$. It is established that if $|x|<1$, then $\lim_{p\to\infty}|x|^p=0$, indicating that the limit is indeed infinite for points on the circumference.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with the concept of circumferences in $\mathbb{R}^2$
- Knowledge of the $\varepsilon-N$ definition of limits
- Basic properties of exponential functions
NEXT STEPS
- Study the properties of limits involving exponential functions
- Explore the concept of $\varepsilon-N$ definitions in more depth
- Investigate the geometric implications of limits in $\mathbb{R}^2$
- Learn about the behavior of functions as they approach infinity
USEFUL FOR
Mathematicians, calculus students, and anyone interested in advanced limit concepts and their applications in geometry.