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Answer the limit and give me the steps and tricks 

The limit problem you provided is:

\[
\lim_{x \to +\infty} \frac{\sqrt{x^3 + 1 - \sin^3(x)}}{\cos^3(x) - \sqrt{x^3 - 1}}
\]

### Step 1: Analyze the leading terms as \( x \) approaches infinity
As \( x \) approaches infinity, the dominant terms inside the square roots will be \( x^3 \) because \( x^3 \) grows much faster than the constants 1 and \(-1\) or the bounded trigonometric functions \(\sin^3(x)\).

Therefore, we can approximate the expressions inside the square roots as:
\[
\sqrt{x^3 + 1 - \sin^3(x)} \approx \sqrt{x^3} = x^{3/2}
\]
\[
\sqrt{x^3 - 1} \approx \sqrt{x^3} = x^{3/2}
\]

### Step 2: Simplify the expression
Now the limit can be rewritten as:
\[
\lim_{x \to +\infty} \frac{x^{3/2}}{\cos^3(x) - x^{3/2}}
\]

### Step 3: Factor out the leading term \( x^{3/2} \) in the denominator
Factor \( x^{3/2} \) from the denominator:
\[
\lim_{x \to +\infty} \frac{x^{3/2}}{x^{3/2} \left(\frac{\cos^3(x)}{x^{3/2}} - 1\right)}
\]

This simplifies to:
\[
\lim_{x \to +\infty} \frac{1}{\frac{\cos^3(x)}{x^{3/2}} - 1}
\]

### Step 4: Consider the behavior of \( \frac{\cos^3(x)}{x^{3/2}} \)
Since \(\cos(x)\) is bounded between \(-1\) and \(1\), \( \cos^3(x) \) is also bounded. Therefore, as \( x \to +\infty \), \( \frac{\cos^3(x)}{x^{3/2}} \) approaches 0.

Thus, the limit simplifies to:
\[
\lim_{x \to +\infty} \frac{1}{-1} = -1
\]

### Final Answer:
\[
\lim_{x \to +\infty} \frac{\sqrt{x^3 + 1 - \sin^3(x)}}{\cos^3(x) - \sqrt{x^3 - 1}} = -1
\]

### Trick:
The key trick is recognizing that the square root terms can be approximated by their leading terms for large \( x \), which greatly simplifies the limit.

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

Here are 5 related questions:
1. How would the limit change if the expression inside the square root was different, like \( \sqrt{x^3 + 2x + \sin(x)} \)?
2. Can you find the limit if the trigonometric function in the numerator was \( \cos^3(x) \) instead of \( \sin^3(x) \)?
3. What would the result be if the limit was taken as \( x \to -\infty \) instead of \( x \to +\infty \)?
4. How does the bounded nature of trigonometric functions affect limits involving them?
5. What techniques can be used to simplify limits involving radicals and trigonometric functions?

**Tip:** Always look for the dominant terms as \( x \) approaches infinity or zero. This simplifies complex expressions and makes it easier to calculate limits.

Learn how to evaluate the limit lim_{x \to +\infty} \frac{\sqrt{x^3 + 1 - \sin^3(x)}}{\cos^3(x) - \sqrt{x^3 - 1}} with detailed steps and tricks. This advanced mathematics problem involves limits, trigonometric functions, and radicals, suitable for advanced mathematics students and enthusiasts.

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