Question 344844
You are exactly correct so far:
{{{(3sqrt(x+h) - 3sqrt(x))/ h}}}
As h approaches 0, this is still of the 0/0 form so we need to do more. With experience you will learn that the next step is to multiply the numerator and denominator by {{{(3sqrt( x+h) + 3sqrt(x))}}}. (You will see why after you see how this simplifies things.)
{{{((3sqrt( x+h) - 3sqrt(x))/ h)((3sqrt( x+h) + 3sqrt(x))/(3sqrt( x+h) + 3sqrt(x)))}}}
In the numerator we can take advantage of the {{{(a+b)(a-b) = a^2 - b^2}}} pattern. (We'll leave the denominator factored for reasons you will see shortly.)
{{{((3sqrt( x+h))^2 - (3sqrt(x))^2)/ ((h)(3sqrt( x+h) + 3sqrt(x)))}}}
Simplifying the numerator:
{{{(9(x+h) - (9x))/ ((h)(3sqrt(x+h) + 3sqrt(x)))}}}
{{{(9x + 9h - 9x)/ ((h)(3sqrt(x+h) + 3sqrt(x)))}}}
{{{9h/((h)(3sqrt(x+h) + 3sqrt(x)))}}}
At this point we will reduce the fraction. (And we can see why we didn't bother multiplying out the denominator.) The h's cancel:
{{{9cross(h)/(cross(h)(3sqrt(x+h) + 3sqrt(x)))}}}
{{{9/(3sqrt(x+h) + 3sqrt(x))}}}
We can factor out a 3 in the denominator:
{{{9/(3(sqrt(x+h) + sqrt(x)))}}}
And cancel a factor of 3:
{{{3/(sqrt(x+h) + sqrt(x))}}}
This is the difference quotient. (As h approaches 0, this fraction is not of the 0/0 form.)