Toward The Largest Gap

$x = (1, 4, 9)$ and $y = (5, 6, 10)$ , so the coordinate gaps are $4, 2, 1$ . The Minkowski distance is

$d_p(x,y) = \left(\sum_i |x_i - y_i|^p\right)^{1/p}$

What is $\lim_{p \to \infty} d_p(x,y)$ , and how does the sequence approach it?

0, since the $1/p$ exponent drives everything toward 1

4, approached from above as the values decrease

7, the Manhattan distance, since that is the largest value in the table

4, approached from below as the values increase with $p$