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We are tasked with solving the limit:

\[
\lim_{x \to \infty} \left( \frac{x^3 + 1}{x^2 + 3} - \frac{x^2 + 1}{x + 2} \right)
\]

### Step 1: Analyzing Each Term Separately

1. **For the first term:** \(\frac{x^3 + 1}{x^2 + 3}\)

   As \(x \to \infty\), the highest degree term in the numerator is \(x^3\) and in the denominator is \(x^2\), so the expression behaves like:

   \[
   \frac{x^3}{x^2} = x
   \]

   Hence, as \(x \to \infty\), this term tends to infinity.

2. **For the second term:** \(\frac{x^2 + 1}{x + 2}\)

   As \(x \to \infty\), the highest degree term in the numerator is \(x^2\) and in the denominator is \(x\), so the expression behaves like:

   \[
   \frac{x^2}{x} = x
   \]

   Hence, as \(x \to \infty\), this term also tends to infinity.

### Step 2: Simplifying the Limit Expression

Since both terms grow asymptotically like \(x\), let’s factor out the dominant powers from both terms to analyze the limit more carefully.

#### For the first term:
\[
\frac{x^3 + 1}{x^2 + 3} = \frac{x^3(1 + \frac{1}{x^3})}{x^2(1 + \frac{3}{x^2})} = x \cdot \frac{1 + \frac{1}{x^3}}{1 + \frac{3}{x^2}}
\]
As \(x \to \infty\), \(\frac{1}{x^3} \to 0\) and \(\frac{3}{x^2} \to 0\), so this term simplifies to:
\[
x \cdot \frac{1 + 0}{1 + 0} = x
\]

#### For the second term:
\[
\frac{x^2 + 1}{x + 2} = \frac{x^2(1 + \frac{1}{x^2})}{x(1 + \frac{2}{x})} = x \cdot \frac{1 + \frac{1}{x^2}}{1 + \frac{2}{x}}
\]
As \(x \to \infty\), \(\frac{1}{x^2} \to 0\) and \(\frac{2}{x} \to 0\), so this term simplifies to:
\[
x \cdot \frac{1 + 0}{1 + 0} = x
\]

### Step 3: Substituting into the Original Limit

Now, substituting these simplified forms back into the original limit expression:

\[
\lim_{x \to \infty} \left( x - x \right) = \lim_{x \to \infty} 0 = 0
\]

### Final Answer:
\[
\lim_{x \to \infty} \left( \frac{x^3 + 1}{x^2 + 3} - \frac{x^2 + 1}{x + 2} \right) = 0
\]

Would you like further clarification on any step?

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Here are five related questions for further exploration:
1. How do you evaluate limits at infinity for rational functions in general?
2. What happens if the degree of the numerator is less than the degree of the denominator in a rational function?
3. What techniques can be used to analyze limits involving more complex expressions?
4. How does factoring dominant terms help simplify limits?
5. How do L'Hôpital's Rule and asymptotic behavior relate to solving limits at infinity?

**Tip:** When evaluating limits at infinity, always compare the degrees of the polynomials in the numerator and denominator to simplify the expression effectively.

Evaluate the limit: lim_{x \to \infty} \left( \frac{x^3 + 1}{x^2 + 3} - \frac{x^2 + 1}{x + 2} \right)

Learn how to solve the limit lim(x→∞) ((x^3 + 1)/(x^2 + 3) - (x^2 + 1)/(x + 2)) with a step-by-step approach. This problem covers limits at infinity, simplifying rational functions, and analyzing asymptotic behavior. Suitable for high school and college calculus students.

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