find lim x->0 3- sqrt x+9/-x

Answer 1

In general, when there is a sum or difference of square-root terms, multiplying both numerator and denominator by the conjugate is a good strategy.

Lim (3- sqrt( x+9))/(-x )
x->0
= Lim (3-sqrt(x+9))(3+sqrt(x+9))/[(-x)(3+sqrt(x+9)]
x->0
= Lim (9-(x+9))/[(-x)(3+sqrt(x+9)]
x->0
= Lim (-x)/[(-x)(3+sqrt(x+9)]
x->0
= Lim 1/(3+sqrt(x+9)
x->0
=1/6