Proof of boundedness of a function Let $|x|<1$ and $f(x)=\displaystyle\frac{e^{\frac{1}{1+x}}}{(|x|-1)^{-2}}.$
Is $f(x)$ bounded?

$|f(x)| \leq e^{\frac{1}{1+x}}$ because the denominator is smaller than 1.
Only for $x=-1$ you will get $\infty$ for the exponential function, but this value is excluded. Hence $f(x)$ is bounded for $|x| < 1$.
