Question

Evaluate the following limit, if it exists:
lim h-->0 ((sqrt9+h)-3/(h)

Answer 1

if f(x) = √x
the limit is f'(9)
f'(x) = 1/(2√x)
f'(9) = 1/6

If you must evaluate the limit, just use the binomial theorem:

(9+h)^(1/2) = 9^(1/2) + (1/2)/1! 9^(-1/2)h + (1/2)(-1/2)/2! 9^(-3/2) h^2 + ...
subtract 3, divide by h, and set h=0