We need to evaluate the limit:

$\lim_{x \to e} (1 + \ln x)$

Step 1: Direct Substitution

Since the function $1 + \ln x$ is continuous for $x > 0$, we can directly substitute $x = e$:

$1 + \ln e$

Step 2: Simplify

We know that:

$\ln e = 1$

Thus:

$1 + 1 = 2$

Conclusion:

$\lim_{x \to e} (1 + \ln x) = 2$

Would you like a more detailed explanation? Here are some related questions:

1. What is the general method for evaluating limits of logarithmic functions?
2. How do you prove the continuity of the function $f(x) = 1 + \ln x$?
3. What happens to $1 + \ln x$ as $x$ approaches 0?
4. How does the derivative of $1 + \ln x$ relate to its limit properties?
5. Can L'Hôpital's Rule be applied to similar logarithmic limits?

Tip: The natural logarithm function $\ln x$ is only defined for $x > 0$, so always check the domain before evaluating limits.