Problem 40 - Algebra and Trigonometry

Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ g(x)=\frac{15 x^{2}}{3 x^{2}+1} $$

Step-by-Step Solution

Answer

The line \(y=5\) is the horizontal asymptote for the given function.

Step 1: Analyze degrees of polynomials

Look at the degrees of the polynomials in the numerator and denominator. Both are of degree 2.

Step 2: Apply the rule for horizontal asymptotes

If the degrees of the polynomials in the numerator and denominator are equal, then the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and the denominator. Here, the leading coefficient for the numerator is 15 and for the denominator is 3.

Step 3: Calculate the horizontal asymptote

The horizontal asymptote would be the ratio of the leading coefficients, that is, \(y = \frac{15}{3} = 5\). Therefore, the line \(y=5\) is the horizontal asymptote for \(g(x) = \frac{15x^2}{3x^2+1}\).