Answer
$a)$ After taking the derivative multiple times, the function repeats itself.
$b)$ $1$
$c)$ see picture
Work Step by Step
$a)$$\lim\limits_{x \to \infty}\frac{x}{\sqrt (x^{2}+1)}$=$\frac{\infty}{\infty}$
$\lim\limits_{x \to \infty}\frac{1}{\frac{1}{2}(x^{2}+1)^{-\frac{1}{2}}(2x)}$
$\lim\limits_{x \to \infty}\frac{\sqrt (x^{2}+1)}{x}$=$\frac{\infty}{\infty}$
$\lim\limits_{x \to \infty}\frac{\frac{1}{2}(x^{2}+1)^{-\frac{1}{2}}(2x)}{1}$
$\lim\limits_{x \to \infty}\frac{x}{\sqrt (x^{2}+1)}$
The limit repeats itself.
$b)$ $\lim\limits_{x \to \infty}\frac{x}{\sqrt (x^{2}+1)}$
$\lim\limits_{x \to \infty}\frac{x(\frac{1}{x})}{\sqrt (x^{2}+1)(\frac{1}{\sqrt x^{2}})}$
$\lim\limits_{x \to \infty}\frac{1}{\sqrt (1+\frac{1}{x^{2}})}$=$\frac{1}{\sqrt 1}$=$1$
$c)$ see picture
