Linear approximation is used when the value needed is close to a known value.
Here, we know sqrt(9)=3.
Let
f(x)=sqrt(x)
f'(x)=1/(2sqrt(x))
Apply linear approximation
f(9.05)=f(9)+f'(x)dx
=sqrt(9)+0.05/(2sqrt(9))
=3+.05/(6)
=3.008333
exact value is 3.008322, estimate is off by .0004%