To find the first-order neighborhood of the function \( f(x) = \frac{1}{x} + \ln(x) \) around a point \( x_0 \), you typically expand the function using a first-order Taylor series expansion around that point. The first-order Taylor expansion is given by:

\[
f(x) \approx f(x_0) + f'(x_0)(x - x_0)
\]

### Step 1: Find \( f(x_0) \)
Let's choose \( x_0 = 1 \) for simplicity (since you didn't specify a point). Plugging \( x_0 = 1 \) into the function:

\[
f(1) = \frac{1}{1} + \ln(1) = 1 + 0 = 1
\]

### Step 2: Find the derivative \( f'(x) \)
First, let's calculate the derivative of the function \( f(x) = \frac{1}{x} + \ln(x) \):

\[
f'(x) = -\frac{1}{x^2} + \frac{1}{x}
\]

### Step 3: Evaluate the derivative at \( x_0 = 1 \)
Now, evaluate this derivative at \( x_0 = 1 \):

\[
f'(1) = -\frac{1}{1^2} + \frac{1}{1} = -1 + 1 = 0
\]

### Step 4: Construct the first-order Taylor expansion
The first-order Taylor expansion around \( x_0 = 1 \) is then:

\[
f(x) \approx f(1) + f'(1)(x - 1)
\]

Substituting the values we found:

\[
f(x) \approx 1 + 0 \cdot (x - 1) = 1
\]

So, the first-order approximation (neighborhood) of \( f(x) = \frac{1}{x} + \ln(x) \) around \( x = 1 \) is simply \( f(x) \approx 1 \).

Would you like more details, or do you have any questions?

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Here are 5 related questions that could further your understanding:

1. How does the second-order Taylor expansion of \( f(x) \) differ from the first-order expansion?
2. What is the geometric interpretation of a first-order Taylor expansion?
3. Can the function \( f(x) = \frac{1}{x} + \ln(x) \) have points of inflection?
4. How would the first-order approximation change if you expanded around a different point, such as \( x_0 = 2 \)?
5. What is the significance of the point where the derivative equals zero in a Taylor series expansion?

**Tip:** Taylor series expansions are powerful tools for approximating functions near a given point. The more terms you include, the more accurate the approximation becomes, especially near the expansion point.

Explore the first-order Taylor series expansion of the function f(x) = 1/x + ln(x) around x = 1. Learn how to approximate functions using Taylor series, including step-by-step calculations and related questions. Ideal for advanced mathematics enthusiasts.

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