Final answer:
The rate of convergence of the sequence (1 - e^h) / h as h approaches 0 is found by applying L'Hôpital's Rule, resulting in the limit equalling -1.
Step-by-step explanation:
The question asks us to find the rate of convergence of the sequence as h approaches 0 for the expression (1 - eh) / h. To address this, we'll look at the limit.
When evaluating the limit of (1 - eh) / h as h approaches 0, we can use L'Hôpital's Rule, which states that if the limit results in an indeterminate form such as 0/0, we can take the derivative of the numerator and the derivative of the denominator and then take the limit of that fraction.
Taking the derivative of the numerator, we have -eh, and the derivative of the denominator is 1. The limit thus becomes:
\[\lim_{h \to 0} \frac{-e^h}{1} = -1\]
This confirms that the original limit of (1 - eh) / h as h approaches 0 is indeed -1, showing the sequence converges to -1 at a rate that can be referred to as 'fast' since the convergence is reached immediately upon evaluation of the limit.