lim x -> - ∞ ((1 - x ^ 3)/(x ^ 2 + 7x)) ^ 5

We are tasked with evaluating the limit:

\[
\lim_{{x \to -\infty}} \left( \frac{1 - x^3}{x^2 + 7x} \right)^5
\]

### Step 1: Focus on the inner expression
Let's first simplify the inner expression:

\[
\frac{1 - x^3}{x^2 + 7x}
\]

As \( x \to -\infty \), the dominant term in the numerator is \( -x^3 \), and in the denominator, the dominant term is \( x^2 \). Hence, the expression simplifies by considering the leading terms:

\[
\frac{1 - x^3}{x^2 + 7x} \sim \frac{-x^3}{x^2} = -x
\]

### Step 2: Applying the limit
Now we know that:

\[
\frac{1 - x^3}{x^2 + 7x} \sim -x
\]

As \( x \to -\infty \), \( -x \to \infty \).

### Step 3: Raising the expression to the 5th power
Next, we raise this simplified expression to the 5th power:

\[
\left( -x \right)^5 = -x^5
\]

Thus, as \( x \to -\infty \), \( -x^5 \to \infty \).

### Conclusion
Therefore, the limit of the given expression is:

\[
\lim_{{x \to -\infty}} \left( \frac{1 - x^3}{x^2 + 7x} \right)^5 = \infty
\]

Would you like further details or explanations on any part of the process?

Here are 5 related questions:
1. What happens to the limit if we change the exponent to \(3\)?
2. How do dominant terms impact limits involving large values of \(x\)?
3. Can we apply L'Hopital's rule to evaluate this limit instead?
4. How do limits behave for polynomial expressions in general?
5. What is the behavior of the expression as \( x \to \infty \) instead of \( x \to -\infty \)?

**Tip**: When simplifying expressions with limits at infinity, always identify and focus on the dominant terms. This helps in approximating the behavior of the function more easily.

Explore how to evaluate the limit of ((1 - x^3)/(x^2 + 7x))^5 as x approaches -∞. This problem uses key concepts from limits, asymptotic behavior, and polynomial dominance. Suitable for advanced high school students or undergraduates.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation