https://file.aiquickdraw.com/mathbot/file/172581366562354gpo65m.jpg

The limit expression provided in the image is:

\[
\lim_{x \to -\infty} \frac{6x^3 + x}{x + e^x}
\]

### Step-by-step solution:

1. **Asymptotic behavior of \(e^x\) when \(x \to -\infty\)**:
   - As \(x \to -\infty\), \(e^x \to 0\). This is because the exponential function \(e^x\) approaches 0 very rapidly as \(x\) becomes more negative.

2. **Analyze the numerator** \(6x^3 + x\):
   - When \(x \to -\infty\), the dominant term in the numerator is \(6x^3\), because cubic terms dominate lower-degree terms. Thus, the numerator behaves like \(6x^3\) as \(x \to -\infty\).

3. **Analyze the denominator** \(x + e^x\):
   - As \(x \to -\infty\), \(e^x\) tends to 0, so the denominator behaves approximately like \(x\). Therefore, the denominator behaves like \(x\) as \(x \to -\infty\).

4. **Simplify the expression**:
   - The limit can now be approximated by:
     \[
     \lim_{x \to -\infty} \frac{6x^3 + x}{x + e^x} \approx \lim_{x \to -\infty} \frac{6x^3}{x} = \lim_{x \to -\infty} 6x^2
     \]
   - Since \(6x^2 \to \infty\) as \(x \to -\infty\), the limit tends to \(+\infty\).

### Conclusion:
\[
\lim_{x \to -\infty} \frac{6x^3 + x}{x + e^x} = +\infty
\]

Let me know if you need any clarifications!

### Related questions:
1. What is the behavior of \(e^x\) as \(x \to +\infty\)?
2. How do we handle limits involving polynomials and exponentials together?
3. What happens when we analyze the limit at \(x \to +\infty\) for this same expression?
4. How can L'Hopital's Rule be applied to evaluate limits involving infinity?
5. What techniques can be used for limits of rational functions?

**Tip:** When dealing with limits involving polynomials and exponentials, always analyze which terms dominate as \(x\) approaches infinity or negative infinity.

Consider the following limit: \(\lim_{x \to -\infty} \frac{6x^3 + x}{x + e^x}\). Is it equal to \(+\infty\), \(-\infty\), or does it approach a finite value?

Learn how to evaluate the limit of \(\frac{6x^3 + x}{x + e^x}\) as \(x\) approaches negative infinity. This detailed step-by-step solution explains the dominance of polynomial terms over exponential functions and uses asymptotic analysis. Ideal for students in Calculus I.

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