Evaluate the limit of (x - 9)/(sqrt(x) - 3) as x approaches 9.

Step-by-step solution

1. 1

   Check the form. At $x=9$ the numerator is $0$ and the denominator is $3-3=0$, so this is $\frac{0}{0}$.

2. 2

   See $x$ as a square. For $x>0$ we have $x=\left(\sqrt{x}\right)^2$, so $x-9=\left(\sqrt{x}\right)^2-3^2$, which is a difference of squares in the variable $\sqrt{x}$.

3. 3

   Factor the numerator. $x-9=\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)$. The first factor is exactly the denominator, so no conjugate multiplication is needed at all.

4. 4

   Cancel. For $x\neq 9$ the quotient reduces to $\sqrt{x}+3$, a function continuous on $x\ge 0$.

5. 5

   Substitute. $\sqrt{9}+3=3+3=6$.

6. 6

   Note the shortcut. Multiplying by the conjugate $\sqrt{x}+3$ produces the same reduction; recognising the difference of squares simply skips a line of work.