lim
x->9
1 1
- - -
3 square root of x
___________________________
x-9
To find the limit as x approaches 9 of the given expression, we can substitute 9 into the expression and simplify.
So, substituting x = 9 into the expression:
[1/(3/sqrt(x))] / (x - 9)
We can simplify this expression step by step:
1. Start by simplifying the denominator (x - 9):
x - 9 = 9 - 9 = 0
2. Simplify the numerator. Since the denominator involves the square root of x, we can't simply substitute 9 into it. However, we can rewrite the expression to remove the square root by squaring both the numerator and denominator. This is equivalent to multiplying by the conjugate of the denominator, which eliminates the square root.
numerator: 1^2 = 1
denominator: (3/sqrt(x))^2 = (3^2)/(sqrt(x))^2 = 9/x
Now the expression becomes:
1 / (9/x)
= x / 9
So, the limit of the given expression as x approaches 9 is x/9.