Question

what is the horizontal asymptote of the graph of $f(x)=\frac{x^{2}+3}{2x^{2}-7x}$?
give your answer in the form $y = a$.
provide your answer below:

Explanation:

Step1: Divide numerator and denominator by \(x^2\)

$$\begin{align*} f(x)&=\frac{\frac{x^{2}}{x^{2}}+\frac{3}{x^{2}}}{\frac{2x^{2}}{x^{2}}-\frac{7x}{x^{2}}}\\ &=\frac{1 + \frac{3}{x^{2}}}{2-\frac{7}{x}} \end{align*}$$

Step2: Find the limit as \(x\to\pm\infty\)

As \(x\to\pm\infty\), \(\frac{3}{x^{2}}\to0\) and \(\frac{7}{x}\to0\)

$$ \lim_{x\to\pm\infty}f(x)=\frac{1 + 0}{2-0}=\frac{1}{2} $$

Answer:

\(y = \frac{1}{2}\)