For the equation y = (x^2 + 2x + 1)/(x^2 + 4x + 3), what is the horizontal asymptote?

To determine the horizontal asymptote of the function given by the equation ( y = \frac{x^2 + 2x + 1}{x^2 + 4x + 3} ), it is important to analyze the degrees of the polynomial in both the numerator and the denominator.

In this case, both the numerator ( x^2 + 2x + 1 ) and the denominator ( x^2 + 4x + 3 ) are quadratic polynomials, meaning their degree is 2. When the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients.

The leading coefficient of both the numerator and the denominator, which are the coefficients of ( x^2 ), is 1. Thus, the horizontal asymptote is determined by:

[

y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{1}{1} = 1.

]

Therefore, the horizontal asymptote of the function is ( y = 1 ). This indicates that as ( x ) approaches either positive or negative infinity, the value of ( y \