Exercise 8e, from your document, asks for the calculation of the following limit:

$\lim_{x \to 1} \frac{e^x - e}{x - 1}$

This is a classic limit problem often solved using L'Hopital's Rule, as it results in an indeterminate form $\frac{0}{0}$.

To apply L'Hopital's Rule, we differentiate the numerator and denominator:

1. The derivative of $e^x - e$ with respect to $x$ is $e^x$.
2. The derivative of $x - 1$ with respect to $x$ is 1.

Thus, applying L'Hopital's Rule:

$\lim_{x \to 1} \frac{e^x - e}{x - 1} = \lim_{x \to 1} \frac{e^x}{1}$

Now, substitute $x = 1$ into the expression:

$\frac{e^1}{1} = e$

Therefore, the value of the limit is $e$, which is approximately 2.718.

Would you like further clarification on this?