Problem

Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by

$$f(x)=\frac{x-1}{\sqrt{x^2+1}}.$$

Note that $f$ is well defined as $x^2+1>0$ for all $x\in\mathbb{R}$.

(a) (10 pts) Please find $\lim_{x\to-\infty}f(x)$. Show your work.

(b) (10 pts) Please find $f'(x)$, the derivative of $f(x)$. Show your work.

(c) (10 pts) Please find the global maximum or minimum value of $f(x)$ for $x\in\mathbb{R}$. Show your work.