Asymptote

An asymptote is a line that a function approaches, but never quite reaches it, in other words, the distance between the function and asymptote tends to zero as the function tends to infinite.

There are three types of asymptotes:

---

Solving a Horizontal Asymptote

Given a function: $f(x)=\frac{2x^2+3}{x^2+1}$one could discover its asymptote using the following rules:

• When the numerator degree is less than the denominator degree, the horizontal asymptote is the x-axis — $y=0$.
• When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

So, in this example, the numerator and denominator degrees are equal — both are squared —, resulting in: ${\lim }_{x\to \pm \infty }\frac{2x^2+3}{x^2+1}=2$

---

Solving a Vertical Asymptote

A vertical asymptote happens when the denominator of a function becomes zero. So, to find it, just calculate the zero of the denominator.

Given the function:

$$f(x)=\frac{5}{x^2}$$

Would result in:

$$\underset{x\to 0}{\lim }\frac{5}{x^2}=\infty$$

---