91Ó°ÊÓ

Let's start by writing down the slope formula, which will be used to calculate the slope of the line: \[m = \frac{y2 - y1}{x2 - x1}\]

Now, we will substitute the coordinates of the two points \((e^2, 6)\) and \((e^{2+r}, 6+r)\) into the slope formula: \[m = \frac{(6 + r) - 6}{(e^{2 + r}) - e^2}\]

Let's simplify the expression by performing arithmetic operations in the numerator and denominator: \[m = \frac{r}{e^{2 + r} - e^2}\]

Since \(r\) is a small positive number, we can take the limit of the expression as \(r\) approaches 0. This will give us an estimate of the slope: \[\lim_{r \to 0} \frac{r}{e^{2+r} - e^2}\] To evaluate this limit, we can use L'Hôpital's rule. To apply L'Hôpital's rule, we must find the derivatives of both the numerator and the denominator with respect to \(r\): \[\frac{d}{dr}(r) = 1\] \[\frac{d}{dr}(e^{2+r} - e^2) = e^{2+r}\] Now, let's apply L'Hôpital's rule: \[\lim_{r \to 0} \frac{1}{e^{2+r}}\] Finally, we can substitute \(r = 0\) into the expression to find the estimated slope: \[\frac{1}{e^2}\] Thus, the estimated slope of the line containing the points \((e^2, 6)\) and \((e^{2+r}, 6+r)\) is \(\frac{1}{e^2}\).