Problem 38 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x)=\frac{15x}{3x^{2}+1}\) is y=0.

Step 1: Identify the type of function

In the function \(f(x)=\frac{15x}{3x^{2}+1}\), the numerator is a polynomial of degree 1 and the denominator is a polynomial of degree 2. This is a rational function.

Step 2: Compare the degree of numerator and denominator

The degree of the numerator (1) is less than the degree of the denominator (2).

Step 3: Find the horizontal asymptote

For a rational function where the degree of the numerator is less than the degree of the denominator, the value of the function approaches zero as x approaches infinity or negative infinity. Hence, the horizontal asymptote is y=0.