valuate the following limits algebraically.
lim
x→e2
√x + ln(x)

To evaluate the limit:

\[
\lim_{x \to e^2} \left( \sqrt{x} + \ln(x) \right)
\]

we substitute \(x = e^2\) directly into the expression, since there are no indeterminate forms.

### Step 1: Substitute \(x = e^2\)
\[
\sqrt{e^2} + \ln(e^2)
\]

### Step 2: Simplify each part
- \(\sqrt{e^2} = e\), because \(\sqrt{e^2} = e^{2/2} = e\).
- \(\ln(e^2) = 2\), because \(\ln(e^2) = 2 \ln(e) = 2\).

### Step 3: Add the results
\[
e + 2
\]

Thus, the limit is:

\[
\lim_{x \to e^2} \left( \sqrt{x} + \ln(x) \right) = e + 2
\]

Would you like further clarification or details?

Here are 5 related questions you can explore:
1. How do you evaluate limits when an indeterminate form appears?
2. What is the derivative of \(\sqrt{x} + \ln(x)\)?
3. How do logarithmic properties help simplify limits?
4. Can L'Hopital's Rule be applied to this type of limit?
5. How do you compute limits involving exponential and logarithmic functions?

**Tip:** Always check for direct substitution first when evaluating limits algebraically to avoid unnecessary steps.

Evaluate the following limits algebraically. lim x→e^2 √x + ln(x)

This guide explains how to evaluate the limit lim x→e^2 √x + ln(x) algebraically using direct substitution. Learn about logarithmic functions, square roots, and limit evaluation techniques suitable for Grades 10-12 students.

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