Chapter 2.3, Problem 44E

To determine

To prove: That for a rational function $r\left(x\right)$ , $\underset{x\to a}{\lim }r\left(x\right)=r\left(a\right)$ for every number a in the domain of r .

Explanation of Solution

Given information:

Given that $r\left(x\right)$ is a rational function.

Consider a rational function as,

  $r\left(x\right)=\frac{a_1{x}^2+b_{1}x+c_1}{a_2{x}^2+b_{2}x+c_2}$

Then $r\left(a\right)=\frac{a_1{a}^2+b_{1}a+c_1}{a_2{a}^2+b_{2}a+c_2}$

Now,

  $\begin{array}{c}\underset{x\to a}{\lim }r\left(x\right)=\underset{x\to a}{\lim }\frac{a_1{x}^2+b_{1}x+c_1}{a_2{x}^2+b_{2}x+c_2}\\ =\frac{a_1{a}^2+b_{1}a+c_1}{a_2{a}^2+b_{2}a+c_2}\\ =r\left(a\right)\end{array}$

Hence, $\underset{x\to a}{\lim }r\left(x\right)=r\left(a\right)$