The function \( f(x) = rac{2x + 1}{x - 3} \) has a horizontal asymptote at \( y = ? \)
["Understanding the Horizontal Asymptote of ( f(x) = \dfrac{2x + 1}{x - 3} )", "When analyzing rational functions like ( f(x) = \dfrac{2x + 1}{x - 3} ), one key concept is the horizontal asymptote—an essential feature that reveals the function’s long-term behavior as ( x ) approaches infinity. In this article, we’ll explore how to determine the horizontal asymptote of ( f(x) = \dfrac{2x + 1}{x - 3} ) and find that it has a horizontal asymptote at ( y = 2 ).", "### What is a Horizontal Asymptote?", "A horizontal asymptote describes a horizontal line ( y = L ) that the graph of a function ( f(x) ) approaches but never crosses as ( x ) grows very large in either the positive or negative direction. To find this asymptote for rational functions, we compare the degrees of the polynomial in the numerator and the denominator.", "### Analyzing the Rational Function", "The function given is:", "[\nf(x) = \dfrac{2x + 1}{x - 3}\n]", "Here, both the numerator and the denominator are linear polynomials (degree 1). When the degree of the numerator and the denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.", "- Leading term in the numerator: ( 2x ) → coefficient: 2\n- Leading term in the denominator: ( x ) → coefficient: 1", "So, the horizontal asymptote is:", "[\ny = \dfrac{2}{1} = 2\n]", "### Why This Rule Works", "As ( x ) approaches ( \infty ) or ( -\infty ), the lower-degree terms become negligible compared to the leading terms. Rewriting ( f(x) ) by dividing numerator and denominator by ( x ):", "[\nf(x) = \dfrac{2x + 1}{x - 3} = \dfrac{2 + \dfrac{1}{x}}{1 - \dfrac{3}{x}}\n]", "As ( x \ o \infty ) or ( x \ o -\infty ), the terms ( \dfrac{1}{x} ) and ( \dfrac{3}{x} ) approach 0. Therefore:", "[\nf(x) \ o \dfrac{2 + 0}{1 - 0} = 2\n]", "This confirms the function approaches and stabilizes at ( y = 2 ).", "### Conclusion", "The rational function ( f(x) = \dfrac{2x + 1}{x - 3} ) has a horizontal asymptote at ( y = 2 ). Recognizing rational asymptotes through degree comparison and limiting behavior helps in predicting function behavior at extreme values, a key skill in calculus and algebra.", "Whether you're solving limits, graphing rational functions, or analyzing real-world models involving ratios, knowing that ( f(x) = \dfrac{2x + 1}{x - 3} ) approaches ( y = 2 ) gives valuable insight into its shape and trends.", "---", "Headline:\nThe Horizontal Asymptote of ( f(x) = \dfrac{2x + 1}{x - 3} ) Is ( y = 2 ) — Here’s Why", "Meta description:\nLearn how to find and understand the horizontal asymptote of the function ( f(x) = \dfrac{2x + 1}{x - 3} ). Discover the value ( y = 2 ) and why it matters in rational function analysis."]
