Question

Let$r\left(x\right)$be a nonzero rational function. Evaluate $\underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)}$.

Short answer

The answer is$1.$

Step-by-step solution

Step 1. Given information.

It is given to evaluate$\underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)}$.

Step 2. Required Value.

Now,

$$\begin{gathered} r(x+1)=\sum _{i=0}^{\infty }\frac{a_i}{b_i}(x+1{)}^i \\ \frac{r(x+1)}{r\left(x\right)}=\frac{\frac{a_i}{b_i}(x+1{)}^{\text{'}}}{\frac{a_i}{b_i}{x}^i} \\ =\frac{(x+1{)}^i}{x^i} \\ ={\left(1+\frac{1}{x}\right)}^i \end{gathered}$$

Applying limit,

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{r(x+1)}{r\left(x\right)}=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{\text{'}} \\ =\underset{x\to \infty }{\lim }(1+0{)}^i \\ =1 \end{gathered}$$