Take a look at the function $f(x)=\tfrac{1}{x}$. In the graph we see that the function consists of two branches . This is because $0$ is not part of the domain of the function and hence, $f(0)$ does not exist. We name a graph consisting of two branches a $\blue{\textbf{hyperbola}}$.

We see that, if $x$ becomes really negative or if $x$ becomes really positive, the graph approaches the $x$-axis really close. However, the function value never becomes equal to $0$. We call the $x$-axis, the line $y=0$, the $\purple{\textbf{horizontal asymptote}}$ of the graph.

We also see that if the $x$-value approaches $0$ from the negative side, the function value becomes arbitrarily negative: it decreases without any bound. On the other hand, if the $x$-value approaches $0$ from the positive side, $f(x)$ becomes arbitrarily positive. We call the $y$-axis, the line $x=0$, the $\green{\textbf{vertical asymptote}}$ of the graph.