which of the following does (4x^2)/(x^2+5000x) approach

To determine what the expression \(\frac{4x^2}{x^2 + 5000x}\) approaches as \(x\) increases, let's analyze its behavior:

### Simplifying the Expression:
We divide both the numerator and the denominator by \(x^2\), the highest power of \(x\) in the expression:

\[
\frac{4x^2}{x^2 + 5000x} = \frac{4}{1 + \frac{5000}{x}}
\]

### As \(x \to \infty\):
- The term \(\frac{5000}{x}\) tends to 0 because \(x\) is becoming very large.
- Therefore, the expression simplifies to:

\[
\frac{4}{1 + 0} = 4
\]

### Conclusion:
As \(x\) approaches infinity, \(\frac{4x^2}{x^2 + 5000x}\) approaches 4.

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. What happens to the expression when \(x \to -\infty\)?
2. How does the behavior change if the coefficient 5000 is replaced with another constant?
3. What would the expression approach if the numerator was \(4x^3\) instead of \(4x^2\)?
4. How do you apply limits to expressions with higher-order polynomials?
5. Can we use L'Hopital's Rule for this limit? Why or why not?

**Tip**: When finding limits of rational functions, dividing by the highest power of \(x\) can help simplify the analysis.

Explore how to find the limit of the rational function (4x^2)/(x^2 + 5000x) as x approaches infinity. This problem focuses on key concepts in limits and asymptotic behavior, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation