limit (x-9)/(sqrt(x))-3
as x-->9
To find the limit of the function (x-9)/(sqrt(x))-3 as x approaches 9, we can use direct substitution. However, if we substitute x = 9 directly into the expression, we get an undefined value in the denominator:
(9-9)/(sqrt(9))-3 = 0/3 - 3 = -3
We can see that direct substitution does not help us in finding the limit because we end up with an indeterminate form.
To evaluate the limit in cases like this, we need to use algebraic manipulation. We can simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator to eliminate the square root.
So, multiplying the numerator and denominator by the conjugate of the denominator (sqrt(x) + 3), we get:
[(x-9)*(sqrt(x) + 3)] / [(sqrt(x) + 3)*(sqrt(x) - 3)]
Expanding and simplifying the expression, we find:
[(x-9)*(sqrt(x) + 3)] / [(sqrt(x))^2 - (3)^2]
= (x-9)*(sqrt(x) + 3) / (x - 9)
= sqrt(x) + 3
Now, we can substitute x = 9 into the simplified expression:
sqrt(9) + 3 = 3 + 3 = 6
Therefore, the limit of the function (x-9)/(sqrt(x))-3 as x approaches 9 is 6.