How do you find the limit of #(sqrtx-3)/(x-9)# as #x->9#?

It should be noted that evaluating the limit in its original form would render the fraction undefinable.

By multiplying the numerator and denominator by the conjugate of the numerator, we can rationalize the number.

We can now determine the limit:

To find the limit of (sqrt(x) - 3)/(x - 9) as x approaches 9, we can use algebraic manipulation and the concept of limits.

First, we notice that the expression is undefined when x = 9, as it would result in division by zero. However, we can still find the limit by simplifying the expression.

We can rewrite the expression as (sqrt(x) - 3)/(x - 9) = [(sqrt(x) - 3)(sqrt(x) + 3)]/[(x - 9)(sqrt(x) + 3)].

Next, we simplify further by canceling out the common factor of (sqrt(x) + 3) in the numerator and denominator, resulting in (sqrt(x) - 3)/(x - 9) = (sqrt(x) + 3)/(x - 9).

Now, we can substitute x = 9 into the simplified expression to find the limit. Plugging in x = 9, we get (sqrt(9) + 3)/(9 - 9) = (3 + 3)/(0) = 6/0.

Since we have a division by zero, the limit does not exist.