\( f(x)=\frac{x^{2}+9 x+18}{x^{2}+6 x+9} \) A. The function has one oblique asymptote, (Type an equation.) B. The function has two different oblique asymptotes. The oblique asymptote as \( x \) approaches \( -\infty \) is \( \square \) and the oblique asymptote as \( x \) approang the graphing strategy and sketch the graph of \( y=f(x) \). (Type is equations.) C. There are no oblique asymptotes.

To analyze the function \( f(x)=\frac{x^{2}+9 x+18}{x^{2}+6 x+9} \), we first need to simplify it. The denominator can be factored as \( (x+3)^2 \). The numerator factors as \( (x+3)(x+6) \). Thus, \( f(x)=\frac{(x+3)(x+6)}{(x+3)^2} \), which simplifies to \( f(x)=\frac{x+6}{x+3} \) for \( x \neq -3 \). A. The oblique asymptote can be found by performing polynomial long division on \( x+6 \) divided by \( x+3 \). The quotient is \( 1 \) and the remainder is \( 3 \), thus the oblique asymptote is \( y=1 \). B. The function has only one oblique asymptote at \( y=1 \) as \( x \) approaches \( \pm\infty \), confirming that \( f(x) \) has the same behavior for both ends as it approaches \( 1 \). This means there are no different oblique asymptotes; the function approaches a horizontal line rather than exhibiting diverse behaviors in either direction. If you want a fun way to remember asymptotes, think of them as the ‘guidelines’ for a dance party! They show where the function is generally heading as the numbers get really big or really small. It's like the path everyone follows even if they occasionally stray! And if you ever want to dive deeper, a classic resource is "Calculus" by James Stewart, where you can dance your way through more complex functions and their behavior!