algebra questions

Question

What is the horizontal asymptote of the rational function f(x) = 3x / (2x - 1)

Answer 1

To clarify, we look at the leading degrees (when put in decending order). Since x = x¹, both the numerator and denominator have a leading degree of 1.

Since they are the same, we divide the leading coefficients.

3/2 = 1 1/2 aka 1.5

Answer 2

Let us rewrite

f(x) = 3x / (2x - 1) as

f(x) = (3x - 1.5 + 1.5) / (2x - 1) = 1.5 + (1.5 / (2x - 1))

As |x| increases, the magnitude of the expression (fraction) in the parentheses becomes smaller and smaller approaching zero, and thus f(x) approaches its horizontal asymptote

y = 1.5