Answer
- Vertical asymptote: $x=1$
- Oblique asymptote: $y=x+1$
Work Step by Step
$$y=\frac{x^2-4}{x-1}$$
- As $x\to1$, $(x-1)$ approaches $0$ and $x^2-4$ approaches $-3\lt0$ , so $(x^2-4)/(x-1)$ will approach $-\infty$. In other words, $$\lim_{x\to1}\frac{x^2-4}{x-1}=-\infty$$
Therefore, the line $x=1$ is the vertical asymptote of the graph.
- There is one more oblique asymptote, but to find it, we need to change the form of the function: $$y=\frac{(x^2-1)-3}{x-1}$$ $$y=\frac{x^2-1}{x-1}-\frac{3}{x-1}$$ $$y=(x+1)-\frac{3}{x-1}$$
As $x\to\pm\infty$, $-3/(x-1)$ approaches $0$, making the function turn into
$$y=x+1$$ which is the oblique asymptote of the graph.
The graph, along with 2 asymptotes, are shown below.
