The formula for a linearized function about a point x=a is :
\(L(x) = f(a)+f'(a)(x-a)\)
Take a = 100
\(f(x)= \sqrt{x}
\\f'(x)= (\sqrt{x} )'= \frac{1}{2 \sqrt{x} }
\\
\\f(100)= \sqrt{100}=10
\\ f'(100)= \frac{1}{2 \sqrt{100} }= \frac{1}{2 \times 10 }= \frac{1}{20 }
\\
\\L(99.8)=10 + \frac{1}{20 }(99.8-100)=10- \frac{0.2}{20}= 9.99\)