Questions: lim (x -> ∞) (4x^5 + 10x^3 + 9x^2 + 2x + 1) / (2x^5 - 3x^4 - 9x + 5)

Solution Steps

To evaluate the limit of the given rational function as \( x \) approaches infinity, we focus on the terms with the highest degree in the numerator and the denominator. The highest degree term in both the numerator and the denominator is \( x^5 \). By dividing every term in the numerator and the denominator by \( x^5 \), we can simplify the expression and find the limit.

We need to evaluate the limit: \[ \lim _{x \rightarrow \infty} \frac{4 x^{5}+10 x^{3}+9 x^{2}+2 x+1}{2 x^{5}-3 x^{4}-9 x+5} \]

As \( x \) approaches infinity, the highest degree terms in both the numerator and the denominator dominate the behavior of the function. The highest degree term in the numerator is \( 4x^5 \) and in the denominator is \( 2x^5 \).

We divide every term in the numerator and the denominator by \( x^5 \): \[ \frac{4 + \frac{10}{x^2} + \frac{9}{x^3} + \frac{2}{x^4} + \frac{1}{x^5}}{2 - \frac{3}{x} - \frac{9}{x^4} + \frac{5}{x^5}} \]

As \( x \) approaches infinity, all terms with \( x \) in the denominator approach zero: \[ \lim _{x \rightarrow \infty} \frac{4 + 0 + 0 + 0 + 0}{2 - 0 - 0 + 0} = \frac{4}{2} = 2 \]

Final Answer