Horizontal Asymptotes and Limits at Infinity

The asymptotes are most commonly encountered in the study of calculus of curves of the form $y = ƒ(x)$. They can be computed using limits and are classified into horizontal, vertical and oblique asymptotes depending on the orientation.

Horizontal asymptotes are horizontal lines that the graph of the function approaches as $x$ tends toward $+ \infty$ or $- \infty$. The horizontal line $y = c$is a horizontal asymptote of the function $y = ƒ(x)$ if $\lim_{x\rightarrow -\infty}f(x)=c$ or $\lim_{x\rightarrow +\infty}f(x)=c$. In the first case, $ƒ(x)$ has $y = c$ as asymptote when $x$ tends toward $- \infty$, and in the second that $ƒ(x)$ has $y = c$ as an asymptote as $x$ tends toward $+ \infty$.

Horizontal asymptote

The graph of a function can have two horizontal asymptotes. An example of such a function would be $y = \arctan(x)$.

Vertical asymptotes are vertical lines (perpendicular to the $x$-axis) near which the function grows without bound. A common example of a vertical asymptote is the case of a rational function at a point $x$ such that the denominator is zero and the numerator is non-zero.

Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches $0$ as $x$ tends toward $+ \infty$ or $- \infty$. More general type of asymptotes can be defined as the oblique asymptote case.

Only open curves that have some infinite branch, can have an asymptote. No closed curve can have an asymptote.