Question

lim (x→∞) (5x³ - 9x⁵)/(3x⁸ + 4x³ - 8) =

Explanation:

Step1: Identify the highest degree term

In the numerator \(5x^3 - 9x^5\), the highest degree term is \(-9x^5\) (degree 5). In the denominator \(3x^8 + 4x^3 - 8\), the highest degree term is \(3x^8\) (degree 8).

Step2: Divide numerator and denominator by the highest degree term in the denominator

Divide each term in the numerator and denominator by \(x^8\):
\[

$$\begin{align*} \lim_{x \to \infty} \frac{\frac{5x^3}{x^8} - \frac{9x^5}{x^8}}{\frac{3x^8}{x^8} + \frac{4x^3}{x^8} - \frac{8}{x^8}}&=\lim_{x \to \infty} \frac{\frac{5}{x^5} - \frac{9}{x^3}}{3 + \frac{4}{x^5} - \frac{8}{x^8}} \end{align*}$$

\]

Step3: Evaluate the limit as \(x \to \infty\)

As \(x \to \infty\), terms with \(x\) in the denominator (\(\frac{5}{x^5}\), \(\frac{9}{x^3}\), \(\frac{4}{x^5}\), \(\frac{8}{x^8}\)) approach 0. So we substitute these limits:
\[
\frac{0 - 0}{3 + 0 - 0}=\frac{0}{3} = 0
\]

Answer:

\(0\)