Q. 90 - Calculus

Question

Use limit rules and the continuity of polynomial functions to prove that every rational function is continuous on its domain.

Step-by-Step Solution

Answer

 A rational function $\mathit{r}\mathbf{=}\frac{\mathbf{p}}{\mathbf{q}}$ is continuous at every point where $\mathit{q}\mathbf{\ne }\mathbf{0}$.

Step 1. Given Information:

Using limit rules and the continuity of power functions.

Step 2. Prove:

Consider any constant function,

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}$

Again, the identity function $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{x}$ are continuous on $\mathrm{ℝ}$

Step 3. Every rational function is continuous in its domain

Now scalar multiples, sums, and products imply that every polynomial is continuous on $\mathbf{ℝ}$

It also follows that a rational function $\mathit{r}\mathbf{=}\frac{\mathbf{p}}{\mathbf{q}}$ is continuous at every point where $\mathit{q}\mathbf{\ne }\mathbf{0}$ .