Let $g(x)=3x+\frac{12}{x}$ for $x>0$.

1. Sketch a graph of $y = g\left(x\right)$, showing that the minimum value of the function occurs when $x = 2$.

2. Calculate the slope of the line tangent to $y = g\left(x\right)$ when $x = 2$.

   Step 1 (b):

   $g^\prime(2)= \lim \limits_{h\to0}\frac{g(2+h)-g(2)}{h}$

   Step 2 (b):

   $\lim \limits_{h\to0}\frac{3(2+h)+\frac{12}{2+h}-\left(3(2)+\frac{12}{2}\right)}{h}$

   Step 3 (b):

   $\lim \limits_{h\to0}\frac{6+3h+\frac{12}{2+h}-12}{h}$

   Step 4 (b):

   $\lim \limits_{h\to0}\frac{-6+3h+\frac{12}{2+h}}{h}\cdot \frac{(2+h)}{(2+h)}$