Calculus of a Single Variable, Chapter 8, 8.7, Section 8.7, Problem 13

Given to solve,
lim_(x->0) (sqrt(25-x^2)-5)/x
upon Rationalizing numerator we get
= lim_(x->0) ((sqrt(25-x^2)-5)/x) ((sqrt(25-x^2)+5)/(sqrt(25-x^2)+5))
=lim_(x->0) (((sqrt(25-x^2)^2-5^2)/(x(sqrt(25-x^2)+5)))
=lim_(x->0) ((((25-x^2)-25)/(x(sqrt(25-x^2)+5)))
=lim_(x->0) ((-x^2)/(x(sqrt(25-x^2)+5)))
=lim_(x->0) ((-x)/((sqrt(25-x^2)+5)))
Now plugging the value of x =0 we get
((-x)/((sqrt(25-x^2)+5)))
= ((-0)/((sqrt(25-0^2)+5)))
=0