lim x->2
[((6-x)^1/2)-2]
_______________
[((3-x)^1/2)-1]
To evaluate the limit of the given expression as x approaches 2, we can use the technique of rationalizing the numerator and denominator.
Let's start by rationalizing the numerator [(6-x)^(1/2) - 2]:
Multiply the numerator and denominator by the conjugate of the numerator, which is [(6-x)^(1/2) + 2]:
[((6-x)^(1/2) - 2) * ((6-x)^(1/2) + 2)]
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[((3-x)^(1/2) - 1) * ((6-x)^(1/2) + 2)]
Now, using the difference of squares formula, we get:
[(6-x) - 4]
___________
[(3-x) - 1]
Simplifying further:
[2-x]
_____
[2]
Since the value of x approaches 2, we can substitute x=2 into the expression:
[2-2]
_____
[2]
Which leads to:
0/2
Finally, simplifying this expression, we get:
0
Therefore, the limit of the given expression as x approaches 2 is 0.