lim x->2

[((6-x)^1/2)-2]
_______________
[((3-x)^1/2)-1]

To evaluate the limit of the given expression, we can plug in the value of x = 2 and simplify. However, this approach could lead to an indeterminate form of 0/0, which means more manipulation is needed.

Let's start by using rationalization to simplify the expression. For the numerator, we multiply both the numerator and denominator by the conjugate of the numerator expression, which is ((6 - x)^(1/2) + 2).

lim x->2 [((6-x)^(1/2) - 2) * ((6 - x)^(1/2) + 2)] / [((3-x)^(1/2) - 1) * ((6 - x)^(1/2) + 2)]

Notice that when we expand the numerator, we have a factor ((6-x)^(1/2))^2 - 2^2, which simplifies to (6 - x) - 4, giving us (2 - x) in the numerator. The term (2 - x) cancels out with one of the factors in the denominator.

lim x->2 [(2 - x) * ((6-x)^(1/2) + 2)] / [((3-x)^(1/2) - 1) * ((6 - x)^(1/2) + 2)]

Next, we can cancel out the common factor of ((6 - x)^1/2 + 2) in both the numerator and denominator:

lim x->2 [(2 - x)] / [((3-x)^(1/2) - 1)]

Now we can plug in x = 2 and evaluate the limit:

[(2 - 2)] / [((3-2)^(1/2) - 1)]
0 / (1 - 1)
0 / 0

At this point, we have obtained an indeterminate form of 0/0, which suggests that further manipulation is necessary. To proceed, we can use L'Hopital's rule, which states that for indeterminate forms, taking the derivative of the numerator and denominator and evaluating the limit again may help us find the answer.

Let's differentiate the numerator and denominator:

d/dx [(2 - x)] / [(3-x)^(1/2) - 1)] = [-1] / [- (3-x)^(1/2) / 2]

After canceling out the negatives, we have:

lim x->2 [1 / (3-x)^(1/2) / 2]
lim x->2 2 / (3-x)^(1/2)

Now we can plug in x = 2 and evaluate the limit:

2 / (3-2)^(1/2)
2 / 1^(1/2)
2 / 1
2

Therefore, the limit of the given expression as x approaches 2 is 2.