Determine the oblique asymptote of the graph of the function. \( g(x)=\frac{x^{2}+5 x-2}{x+3} \)

To find the oblique asymptote of the function \( g(x) = \frac{x^{2} + 5x - 2}{x + 3} \), we can perform polynomial long division. 1. Divide \( x^2 \) (the leading term of the numerator) by \( x \) (the leading term of the denominator), which gives us \( x \). 2. Multiply \( x \) by \( x + 3 \) to get \( x^2 + 3x \). 3. Subtract \( x^2 + 3x \) from the original numerator \( x^2 + 5x - 2 \): \[ (x^2 + 5x - 2) - (x^2 + 3x) = 2x - 2 \] 4. Now, divide \( 2x \) by \( x \), yielding \( 2 \). 5. Multiply \( 2 \) by \( x + 3 \) to get \( 2x + 6 \). 6. Subtract: \[ (2x - 2) - (2x + 6) = -8 \] Now we have: \[ g(x) = x + 2 + \frac{-8}{x + 3} \] As \( x \) approaches infinity, the term \( \frac{-8}{x + 3} \) approaches zero. Therefore, the oblique asymptote is: \[ y = x + 2 \]