find the limit x->infinity 3x^3-5x^2+7/(8+2x+5x^3)

We are tasked with finding the limit of the following function as \( x \to \infty \):

\[
\lim_{{x \to \infty}} \frac{3x^3 - 5x^2 + 7}{8 + 2x + 5x^3}
\]

### Step-by-step process:

1. **Identify the dominant terms**: 
   As \( x \to \infty \), the terms with the highest powers of \( x \) in both the numerator and denominator will dominate. 
   - In the numerator: \( 3x^3 \) dominates over \( -5x^2 \) and \( +7 \).
   - In the denominator: \( 5x^3 \) dominates over \( 8 \) and \( 2x \).

2. **Factor out the highest power of \( x \)** from both the numerator and denominator**:
   To simplify, we factor out \( x^3 \) from both the numerator and denominator:
   
   \[
   \frac{3x^3 - 5x^2 + 7}{8 + 2x + 5x^3} = \frac{x^3(3 - \frac{5}{x} + \frac{7}{x^3})}{x^3(\frac{8}{x^3} + \frac{2}{x^2} + 5)}
   \]

3. **Simplify the expression**:
   After factoring, cancel out \( x^3 \) from the numerator and denominator:

   \[
   = \frac{3 - \frac{5}{x} + \frac{7}{x^3}}{\frac{8}{x^3} + \frac{2}{x^2} + 5}
   \]

4. **Take the limit as \( x \to \infty \)**:
   As \( x \to \infty \):
   - \( \frac{5}{x} \to 0 \), \( \frac{7}{x^3} \to 0 \), \( \frac{8}{x^3} \to 0 \), and \( \frac{2}{x^2} \to 0 \).
   - So, the expression simplifies to:

   \[
   = \frac{3}{5}
   \]

Thus, the limit is:

\[
\boxed{\frac{3}{5}}
\]

Would you like further details on any step, or do you have any questions?

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Here are 5 related questions to consider:

1. How do we find limits when \( x \to 0 \) instead of \( x \to \infty \)?
2. What happens if the highest degree term is in the denominator?
3. How do limits work for trigonometric functions as \( x \to \infty \)?
4. What is the significance of factoring out the highest power in limits involving polynomials?
5. Can you calculate limits involving exponential functions, like \( e^x \), as \( x \to \infty \)?

**Tip**: When calculating limits at infinity for rational functions, focus on the highest-degree terms in the numerator and denominator!

Discover how to find the limit as x approaches infinity for the rational function (3x^3 - 5x^2 + 7) / (8 + 2x + 5x^3). This solution covers key calculus concepts like identifying dominant terms and applying limit properties. Suitable for high school and early college students.

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