Answer
See the details below.
Work Step by Step
We are given the function $ f(x)=\tan \left(\frac{1}{x^2+1}\right)$. Since $\frac{1}{x^2+1}$ is a quotient of polynomials, and $1+x^2$ can not be zero for any $ x\in R $, then by Theorem 2, the function $\frac{1}{x^2+1}$ is continuous. Now, by Theorem 3, the function
$$ f(x)=\tan \left(\frac{1}{x^2+1}\right)$$
is continuous.
