Answer
The equation of the normal line is
$$y_n=-4x+18.$$
Work Step by Step
Step 1: Find the slope of the tangent at $P(4,2)$. By definition of the slope at $a=4$ we get
$$m=\lim_{h\to 0}\frac{f(4+h)-f(4)}{h}=\lim_{h\to0}\frac{\sqrt{4+h}-\sqrt{4}}{h}=\lim_{h\to0}\frac{\sqrt{4+h}-2}{h}=\lim_{h\to0}\frac{\sqrt{4+h}-2}{h}\cdot\frac{\sqrt{4+h}+2}{\sqrt{4+h}+2}=\lim_{h\to0}\frac{\sqrt{4+h}^2-2^2}{h(\sqrt{4+h}+2)}=\lim_{h\to0}\frac{4+h-4}{h(\sqrt{4+h}+2)}=\lim_{h\to0}\frac{h}{h(\sqrt{4+h}+2)}=\lim_{h\to0}\frac{1}{\sqrt{4+h}+2}=\frac{1}{\sqrt{4+0}+2}=\frac{1}{4}.$$
Step 2: When we have the slope, the equation of the tangent at $P(4,2)$ is given by $y-2=m(x-4)$. Putting $m=\frac{1}{4}$:
$$y-2=\frac{1}{4}(x-4)=\frac{1}{4}x-1$$
which gives $$y=\frac{1}{4}x+1.$$
Step 3: The equation of the normal line at the point $P(a,f(a))$ when we have the equation of the tangent $y=mx+n$ is given by $y_n=cx+d$ such that $m\cdot c=-1$ and $c\cdot a+d=f(a)$. For $a=4$, $f(a)=2$ and $m=\frac{1}{4}$ we get
$$\frac{1}{4}\cdot c=-1,\quad c\cdot 4+d=2.$$
From the first equation we get $c=-4$. Putting this into the second equation we get
$$-4\cdot4+d=2\Rightarrow -16+d=2\Rightarrow d=18.$$
Finally we have
$$y_n=-4x+18.$$
