factor $\displaystyle x^2$ out of the square root. you get
$\displaystyle \frac {14x - 5}{\sqrt{x^2} \sqrt{1 + \frac {14}x - \frac 5{x^2}} + x} = \frac {14x - 5}{|x| \sqrt{1 + \frac {14}x - \frac 5{x^2}} + x}$
now in this case, since $\displaystyle x \to \infty$, we have $\displaystyle x$ is positive, and hence, $\displaystyle |x| = x$, so we get
$\displaystyle \frac {14x - 5}{x \sqrt{1 + \frac {14}x - \frac 5{x^2}} + x}$
now divide the top and bottom by $\displaystyle x$ and take the limit