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The given problem is:

\[
\lim_{x \to -\infty} \left( x^6 + x^7 \right)
\]

### Step-by-step solution:

1. **Factor out the highest power of \(x\):**
   \[
   x^6 + x^7 = x^6(1 + x)
   \]
   Here, we factored out \(x^6\), which is the lower degree term, from both terms inside the limit.

2. **Analyze the limit as \(x \to -\infty\):**
   - When \(x \to -\infty\), the term \(1 + x\) tends to a large negative value because \(x\) dominates.
   - Since \(x^6\) is a positive term (even powers of \(x\) are always positive), multiplying by the large negative \(x\) will result in a large negative value.

Thus, the overall limit is:

\[
\lim_{x \to -\infty} \left( x^6 + x^7 \right) = -\infty
\]

### Final Answer:
\[
- \infty
\]

Would you like further details on this solution?

Here are 5 related questions to consider:
1. How would the limit change if the power of \(x^7\) was even?
2. What is the limit of \(x^6 + x^7\) as \(x \to +\infty\)?
3. How do odd and even powers of \(x\) affect limits as \(x \to -\infty\)?
4. How would the behavior of this limit change if the expression was \(x^6 - x^7\)?
5. Can you solve a similar limit with higher degree polynomials, such as \(x^8 + x^9\)?

**Tip:** When solving limits at infinity, always factor out the highest power of \(x\) to simplify the expression.

Find the limit: lim(x → -∞) (x^6 + x^7). If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.

Learn how to find the limit of the polynomial function x^6 + x^7 as x approaches negative infinity. This problem involves key calculus concepts such as limits and polynomial functions, emphasizing the dominance of the leading term.

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