Q. 100 - Calculus

Question

Use the composition rule for limits and the fact that $\tan x$ is continuous on its domain to prove that ${\tan }^{-1}x$ is continuous everywhere.

Step-by-Step Solution

Answer

It is proved that ${\tan }^{-1}x$  is continuous on its domain. 

Step 1. Given Information

We are given that $\tan x$ is continous. 

Step 2. Proving the statement

Using composition rule,

If $x=c$ is in the domain of $\tan x$.

So, $\underset{x\to c}{\lim }\tan x=\tan c$ is the composition given by,

$$\begin{gathered} \underset{x\to c}{\lim }\tan \left({\tan }^{-1}x\right)=\tan \left(\underset{x\to c}{\lim }{\tan }^{-1}x\right) \\ =\tan \left({\tan }^{-1}c\right) \end{gathered}$$

Hence, ${\tan }^{-1}c$ is continous.