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We are tasked with finding the limit:

\[
L = \lim_{x \to \infty} \left( \sqrt[3]{x^3 + 2x^2 + 3x - 4} - \sqrt[3]{x^3 - 2x^2 + 2x - 1} \right)
\]

### Step 1: Factor out dominant term from cube roots

As \(x \to \infty\), the dominant term inside each cube root is \(x^3\). We can factor this term from the cube roots to simplify:

\[
\sqrt[3]{x^3 + 2x^2 + 3x - 4} = x \cdot \sqrt[3]{1 + \frac{2}{x} + \frac{3}{x^2} - \frac{4}{x^3}}
\]
\[
\sqrt[3]{x^3 - 2x^2 + 2x - 1} = x \cdot \sqrt[3]{1 - \frac{2}{x} + \frac{2}{x^2} - \frac{1}{x^3}}
\]

### Step 2: Apply first-order approximation

For large \(x\), we can approximate the cube root expressions using a binomial expansion. Since \(x\) is large, terms like \(\frac{2}{x}\), \(\frac{3}{x^2}\), etc., become small, and we expand the cube root as follows:

\[
\sqrt[3]{1 + u} \approx 1 + \frac{u}{3} \quad \text{for small } u
\]

Using this approximation, we have:
\[
\sqrt[3]{1 + \frac{2}{x} + \frac{3}{x^2} - \frac{4}{x^3}} \approx 1 + \frac{2}{3x}
\]
\[
\sqrt[3]{1 - \frac{2}{x} + \frac{2}{x^2} - \frac{1}{x^3}} \approx 1 - \frac{2}{3x}
\]

### Step 3: Subtract the two approximations

Now, subtract the two approximations:
\[
\left( 1 + \frac{2}{3x} \right) - \left( 1 - \frac{2}{3x} \right) = \frac{2}{3x} + \frac{2}{3x} = \frac{4}{3x}
\]

Thus, for large \(x\), the difference between the two cube roots behaves as:
\[
\sqrt[3]{x^3 + 2x^2 + 3x - 4} - \sqrt[3]{x^3 - 2x^2 + 2x - 1} \approx \frac{4}{3x}
\]

### Step 4: Take the limit as \(x \to \infty\)

Finally, as \(x \to \infty\), \(\frac{4}{3x} \to 0\). Therefore, the limit of the expression is:

\[
L = 0
\]

However, the approximation suggests an additional term (specifically \(x\) behavior at infinity). Therefore, we should correct and reanalyze based on the understanding that

Calculate the limit: L = lim(x → ∞) (∛(x^3 + 2x^2 + 3x - 4) - ∛(x^3 - 2x^2 + 2x - 1))

Find the limit of the expression L = lim(x → ∞) (∛(x^3 + 2x^2 + 3x - 4) - ∛(x^3 - 2x^2 + 2x - 1)). This problem involves limits, binomial approximation, and cubic roots. Suitable for university-level calculus.

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