Q. 14 - Calculus

Question

Suppose$\frac{p\left(x\right)}{q\left(x\right)}$ is a rational function with deg$\left(p\right(x\left)\right)$$=n$and$deg\left(q\right(x\left)\right)$$=m,$if $n<m$then,$\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?,$ if $n=m,$ then $\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?,$and if   $n>m, then \underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?.$

Step-by-Step Solution

Answer

Suppose $\frac{p\left(x\right)}{q\left(x\right)}$is a rational function with $deg\left(p\right(x\left)\right)=n and deg( q(x\left)\right)=m$, if $n<m, then \underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=0, if n=m, then \underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=\frac{a_n}{b_m}$, and if $n>m$, then $\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=not defined.$

Step 1. Given Information.

The given statement is ,

Suppose $\frac{p\left(x\right)}{q\left(x\right)}$is a rational function with $$\begin{gathered} deg\left(p\right(x\left)\right)=n\text{ and }deg\left(q\right(x\left)\right)=m\text{, if }n<m\text{ then, }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?\text{, if }n=m,\text{ then }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?, \\ \text{ and if }n>m\text{, then }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=? \end{gathered}$$

Step 2. Fill the blanks.

Suppose $\frac{p\left(x\right)}{q\left(x\right)}$is a rational function with $$\begin{gathered} deg\left(p\right(x\left)\right)=n\text{ and }deg\left(q\right(x\left)\right)=m\text{, if }n<m,\text{ then }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=0,\text{ if }n=m, \\ \text{ then }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=\frac{a_n}{b_m}\text{, and if }n>m,\text{ then }\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=\text{ not de fined } \end{gathered}$$