Answer:
1/27
Explanation:
Straightforward evaluation of the expression at x=9 gives 0/0, indicating that L'Hôpital's Rule should be used to find the limit. The next step is to find the ratio of the derivatives of numerator and denominator at the limit point.
Let's call the parts of the expression ...
n = 3 -√x
d = 27 -√(x^3)
Then ...
dn/dx = (-1/2)x^(-1/2) . . . . = -1/6 at x=9
dd/dx = (-3/2)x^(1/2) . . . . = -9/2 at x=9
The ratio of these is ...
(-1/6)/(-9/2) = (1/6)(2/9) = 1/27
The limit as x approaches 9 of the given rational expression is 1/27.