Questions: Find the limit. lim as x approaches infinity of (x^2 - 9x + 9) / (x^3 + 3x^2 + 11)

Solution Steps

To find the limit of the given function as \( x \) approaches infinity, we need to analyze the degrees of the polynomials in the numerator and the denominator. The highest degree term in the numerator is \( x^2 \) and in the denominator is \( x^3 \). As \( x \) approaches infinity, the lower degree terms become negligible. Therefore, the limit can be approximated by the ratio of the leading terms.

To find the limit as \( x \) approaches infinity, we first identify the leading terms in the numerator and the denominator. The leading term in the numerator is \( x^2 \) and in the denominator is \( x^3 \).

We simplify the given expression by focusing on the leading terms: \[ \lim_{x \to \infty} \frac{x^2 - 9x + 9}{x^3 + 3x^2 + 11} \approx \lim_{x \to \infty} \frac{x^2}{x^3} \]

Simplify the fraction: \[ \frac{x^2}{x^3} = \frac{1}{x} \]

Evaluate the limit of the simplified expression as \( x \) approaches infinity: \[ \lim_{x \to \infty} \frac{1}{x} = 0 \]

Final Answer