Finding the Horizontal Asymptotes of a Function

Find the horizontal asymptotes, if any, of \(f(x)=\dfrac{3x-2}{4x-1}.\)

Solution We examine the two limits at infinity: \( \lim\limits_{x\rightarrow -\infty }\dfrac{3x-2}{4x-1}\) and \( \lim\limits_{x\rightarrow \infty }\dfrac{3x-2}{4x-1}.\)

Since \(\lim\limits_{x\rightarrow -\infty }\dfrac{3x-2}{4x-1}=\dfrac{3}{4},\) the line \(y=\dfrac{3}{4}\) is a horizontal asymptote of the graph of \(f\) for \( x\) unbounded in the negative direction.

Since \(\lim\limits_{x\rightarrow \infty }\dfrac{3x-2}{4x-1}=\dfrac{3}{4},\) the line \(y=\dfrac{3}{4}\) is a horizontal asymptote of the graph of \(f\) for \( x\) unbounded in the positive direction.