Use the Limit Properties to find the following limit, if it exists. \( \lim _{x \rightarrow \infty} \frac{x^{2}+4}{x^{3}+9} \)

To find the limit \( \lim _{x \rightarrow \infty} \frac{x^{2}+4}{x^{3}+9} \), we can apply the properties of limits and simplify the expression. As \( x \) approaches infinity, we can focus on the leading terms of the numerator and the denominator. Dividing both the numerator and the denominator by \( x^3 \), the highest power of \( x \) in the denominator, gives us: \[ \lim _{x \rightarrow \infty} \frac{\frac{x^{2}}{x^{3}} + \frac{4}{x^{3}}}{\frac{x^{3}}{x^{3}} + \frac{9}{x^{3}}} = \lim _{x \rightarrow \infty} \frac{\frac{1}{x} + \frac{4}{x^{3}}}{1 + \frac{9}{x^{3}}} \] As \( x \) approaches infinity, both \( \frac{1}{x} \) and \( \frac{4}{x^{3}} \) approach 0, while \( \frac{9}{x^{3}} \) also approaches 0. Thus, we simplify: \[ = \frac{0 + 0}{1 + 0} = \frac{0}{1} = 0 \] Therefore, the limit is: \[ \lim _{x \rightarrow \infty} \frac{x^{2}+4}{x^{3}+9} = 0 \]