Question

Do polynomial functions have asymptotes? If yes, how do you find them?

Answer 1

Hint: In this problem, we have to find whether the polynomial functions have asymptotes. We know that the polynomial functions have asymptotes in some cases. We can find them using the limit functions to prove it. We know that the asymptotes are of two types, the vertical one and the horizontal one.

Complete step-by-step answer:
We should know that the only polynomial functions that have asymptotes are the ones whose degree is 0 (horizontal asymptote) and 1 (oblique asymptote). i.e. functions whose graphs are straight lines.
Therefore, we can say that a polynomial function has an asymptote.
We can now find the possible asymptotes.
The only boundary points of the domain are $$+\mathrm{\infty }$$ and $$-\mathrm{\infty }$$, because polynomials can always be defined on the whole real line. So, we cannot find the vertical asymptotes (the only ones that can be found in correspondence of finite boundary points).
For $$f:\mathbb{R}\to \mathbb{R}$$ a general function, the conditions to have an asymptote at $$+\mathrm{\infty }$$ is that the following limits exist and are finite.
$$m:={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\displaystyle \frac{f\left(x\right)}{x}},q:={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}[f\left(x\right)-mx]}}$$
If what stated here is true, then is the line given by the equation $$y=mx+q$$.
Let us show that m exists when f is a general polynomial p of degree n and it is finite if and only if $$n\le 1$$.
For n = 0, we have $$p\left(x\right)=c_0$$ and
$$m:={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\displaystyle \frac{p\left(x\right)}{x}},{\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\displaystyle \frac{c_0}{x}}=0}}$$ for all $$c_0\in \mathbb{R}$$
For n = 1, we have $$p\left(x\right)=c_0+c_{1}x$$ and
$$m:={\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\displaystyle \frac{p\left(x\right)}{x}},{\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\displaystyle \frac{c_0+c_{1}x}{x}}=c_1}}$$, $$\mathrm{\forall }{c}_0,c_1\in \mathbb{R}$$
For, $$n>1$$ all the denominators of the sum will be positive power of x, so their limit is zero.
So, asymptotes can exist only for n = 0, 1, because in the other cases the limit m diverges.

Note: We can now end the proof and show that asymptotes at infinities exist if and only if the degree of the polynomial function is less or equal to 1. Their equation coincides with the equation of the function. We should know that the only polynomial functions that have asymptotes are the ones whose degree is 0 (horizontal asymptote) and 1 (oblique asymptote). i.e. functions whose graphs are straight lines.