Answer
a. See graph.
b. No vertical tangent.
Work Step by Step
a. See graph; it appears to have a vertical tangent at $x=4$
b. Evaluate the limit at $x=4$: $\lim_{h\to0^-}\frac{f(4+h)-f(4)}{h}=\lim_{h\to0^-}\frac{\sqrt {|4-4-h|}-\sqrt {|4-4|}}{h}=\lim_{h\to0^-}\frac{\sqrt {|h|}}{-|h|}=\lim_{h\to0^-}\frac{1}{-\sqrt {|h|}}=-\infty$ and
$\lim_{h\to0^+}\frac{f(4+h)-f(4)}{h}=\lim_{h\to0^+}\frac{\sqrt {|4-4-h|}-\sqrt {|4-4|}}{h}=\lim_{h\to0^+}\frac{\sqrt {h}}{h}=\lim_{h\to0^+}\frac{1}{\sqrt {h}}=\infty$
thus the function does not have a vertical tangent at $x=4$.
