Limit of Rational Function as x Approaches Infinity for (7x^9 - 2x^5 + 9) / (2x^3 + 5x^2

Math Problem Statement

Find the limit as x approaches infinity for the expression: (7x^9 - 2x^5 + 9) / (2x^3 + 5x^2 - 7).

We are tasked with solving the limit of the given expression as $x \to \infty$ :

$\lim_{x \to \infty} \frac{7x^9 - 2x^5 + 9}{2x^3 + 5x^2 - 7}$

Determine the dominant term in both the numerator and the denominator:
- In the numerator, the highest degree term is $7x^9$ .
- In the denominator, the highest degree term is $2x^3$ .
Divide every term in both the numerator and the denominator by $x^9$ (since $x^9$ is the highest degree of the polynomial in the numerator):

= \lim_{x \to \infty} \frac{7 - \frac{2}{x^4} + \frac{9}{x^9}}{\frac{2}{x^6} + \frac{5}{x^7} - \frac{7}{x^9}}$$ 3. **Analyze the behavior as $$x \to \infty$$**: - Terms like $$\frac{2}{x^4}$$, $$\frac{9}{x^9}$$, $$\frac{2}{x^6}$$, etc., approach 0 as $$x \to \infty$$. 4. **Simplify the limit**: - The numerator simplifies to $$7$$ as all other terms go to 0. - The denominator simplifies to 0 because all terms go to 0. So, the expression becomes: $$\lim_{x \to \infty} \frac{7}{0}$$ Since dividing by zero leads to an undefined or infinite result, the limit is: $$\boxed{\infty}$$ ### Conclusion: The limit of the given expression as $$x \to \infty$$ is $$\infty$$. Would you like more details on any step or further clarifications? ### Related questions: 1. What would happen if the highest degree term in the denominator were higher than in the numerator? 2. How does the behavior of limits change if we were looking at $$x \to -\infty$$? 3. Can L'Hopital's Rule be applied to this limit? 4. What are the steps to analyze a rational function's behavior at $$x = \infty$$? 5. How do asymptotes relate to limits at infinity in rational functions? ### Tip: Always check the degrees of the polynomial terms in limits involving rational functions, as they usually determine the behavior at infinity.

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Limits
Polynomials
Rational Functions

lim(x → ∞) f(x)/g(x) where f(x) and g(x) are polynomials
Dominant term approach in limits

Degree comparison of polynomials
Limits of rational functions as x approaches infinity

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