Question

Find the vertical and horizontal asymptotes of the graph of f(x)=3x+1/x^2-4. plot the graph of function and asymptotes

Answer 1

$\displaystyle f(x) = \frac{3x + 1}{x^2 - 4} \\ \textsf{Vertical assymtote}:\,\, x^2 - 4 = 0\\ x = \pm 2 \\ \begin{aligned} \lim_{x \to \infty} f(x) &= \lim_{x \to \infty}\frac{3x + 1}{x^2 - 4}\\ &= \lim_{x \to \infty}\frac{\frac{3}{x} + \frac{1}{x^2}}{1 - \frac{4}{x^2}} &= 0 \end{aligned} \\ \textsf{Horizontal assymtote}:\,\, y = 0\\$