To find the smallest value of 'k' for the following equation

Let $f :\mathbb{R}^+ \to \mathbb{R} \quad $such that$\quad f(x) = \sqrt{x^2 + ax} - \sqrt{x^2+bx}$

Notice that $f(x)$ is increasing when $a>b$ , $f(x) = 0$ when $a=b$ and $f(x)$ is decreasing when $a<b$.

Now when $a>b$ $\lim_{x \to \infty} f(x) = 5$

Hence $\sqrt{x^2 + ax} - \sqrt{x^2+bx} < 5 \qquad \forall a,b \in \mathbb{R}^+$

An elementary way ($x>0$): $$f(x) = \sqrt{x^2 + ax} - \sqrt{x^2+bx} = \frac{x^2 + ax - (x^2+bx)}{\sqrt{x^2 + ax} + \sqrt{x^2+bx}} =\frac{(a - b)x}{x\sqrt{1 + \frac{a}{x}} + x\sqrt{1 + \frac{b}{x}}}= \frac{(a - b)}{\sqrt{1 + \frac{a}{x}} + \sqrt{1 + \frac{b}{x}}} < \frac{a-b}{2}= 5$$