Questions: lim (x, y) -> (9,9) (x^2 - xy) / (sqrt(x) - sqrt(y)) =

Transcript text: \[ \lim _{(x, y) \rightarrow(9,9)} \frac{x^{2}-x y}{\sqrt{x}-\sqrt{y}}= \]

Solution

Solution Steps

To solve the given limit problem, we can use algebraic manipulation to simplify the expression. The expression involves a difference of squares in the numerator and a difference of square roots in the denominator. We can factor the numerator and rationalize the denominator to simplify the expression and then evaluate the limit as \( (x, y) \rightarrow (9, 9) \).

Step 1: Simplify the Expression

The given expression is:

\[ \frac{x^2 - xy}{\sqrt{x} - \sqrt{y}} \]

To simplify, notice that the numerator \(x^2 - xy\) can be factored as \(x(x - y)\).

Step 2: Rationalize the Denominator

The denominator \(\sqrt{x} - \sqrt{y}\) can be rationalized by multiplying the numerator and the denominator by the conjugate \(\sqrt{x} + \sqrt{y}\):

\[ \frac{x(x - y)}{(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y})} = \frac{x(x - y)}{x - y} \]

Step 3: Cancel Common Terms

Since \(x \neq y\) in the limit process, we can cancel the common term \(x - y\) from the numerator and the denominator:

\[ x \]

Step 4: Evaluate the Limit

Now, evaluate the limit as \((x, y) \rightarrow (9, 9)\):

\[ \lim_{(x, y) \rightarrow (9, 9)} x = 9 \]