https://r2.erweima.ai/mathbot/image/1726844210144dl4farsj.jpg

The limit in the given image is:

\[
\lim_{x \to 0^+} \frac{\ln(x + \arccos^3(x)) - \ln(x)}{x^2}
\]

We will approach the solution step by step, using series expansions and applying L'Hopital's Rule if necessary. Here's how we can tackle it:

### Step 1: Expand functions near \( x \to 0^+ \)
We will first expand both the logarithmic function and the \( \arccos(x) \) function around \( x = 0 \).

- The small \( x \) expansion of \( \arccos(x) \) around 0 is:
  \[
  \arccos(x) \approx \frac{\pi}{2} - x + O(x^3)
  \]
  Thus, \( \arccos^3(x) \) can be approximated as:
  \[
  \arccos^3(x) \approx \left(\frac{\pi}{2}\right)^3 - 3\left(\frac{\pi}{2}\right)^2 x + O(x^3)
  \]

- Now, we expand \( \ln(x + \arccos^3(x)) \) around \( x = 0 \):
  Since \( \arccos^3(x) \) dominates for small \( x \), this logarithmic term behaves as:
  \[
  \ln(x + \arccos^3(x)) \approx \ln\left(\left(\frac{\pi}{2}\right)^3\right) + \frac{x}{\left(\frac{\pi}{2}\right)^3} + O(x^2)
  \]

### Step 2: Simplify the numerator
The difference in the numerator \( \ln(x + \arccos^3(x)) - \ln(x) \) can be handled using the logarithmic identity:
\[
\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)
\]
Thus,
\[
\ln(x + \arccos^3(x)) - \ln(x) = \ln\left(1 + \frac{\arccos^3(x)}{x}\right)
\]
For small \( x \), this becomes approximately:
\[
\ln\left(1 + \frac{\left(\frac{\pi}{2}\right)^3}{x}\right) \approx \ln\left(\frac{\left(\frac{\pi}{2}\right)^3}{x}\right)
\]
This simplifies to:
\[
\ln\left(\frac{\left(\frac{\pi}{2}\right)^3}{x}\right) = \ln\left(\left(\frac{\pi}{2}\right)^3\right) - \ln(x)
\]
Thus, the numerator approximately becomes:
\[
\ln\left(\frac{\pi^3}{8}\right)
\]

### Step 3: Apply the limit
Now, we compute the limit of the overall expression:
\[
\lim_{x \to 0^+} \frac{\ln\left(\frac{\pi^3}{8}\right)}{x^2}
\]
Since \( \ln\left(\frac{\pi^3}{8}\right) \) is a constant and the denominator goes to 0 as \( x^2 \to 0 \), the overall limit becomes:

\[
\boxed{\infty}
\]

This is the final result.

---

Would you like more details on any of the steps, or do you have any further questions?

### Related Questions:
1. How do you expand logarithmic functions around a point?
2. What is L'Hopital's Rule and when should it be applied?
3. How can we apply Taylor series expansions to simplify limits?
4. What are common approximations for trigonometric functions near 0?
5. How does the behavior of limits depend on small perturbations?

### Tip:
When handling logarithmic expressions involving sums, the small \( x \) expansion or series expansion can significantly simplify the problem by turning complex expressions into more manageable terms.

lim_{x→0^+} (ln(x + arccos^3(x)) - ln(x)) / x^2

Solve the limit problem: lim_{x→0^+} (ln(x + arccos^3(x)) - ln(x)) / x^2 using series expansion and logarithmic properties. This solution covers key calculus concepts like L'Hopital's Rule and expansions for small x.

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