When working with rational functions, it's essential to understand that the limit of a function can often be evaluated directly unless the denominator equals zero. In cases where the denominator is zero, we can still find the limit by factoring and canceling common factors. However, when the rational function includes radicals, the process requires a different approach. Instead of factoring, we multiply both the numerator and denominator by the conjugate of the radical expression in the denominator.

For example, consider the limit of the function \(\frac{x - 2}{\sqrt{x} - \sqrt{2}}\) as \(x\) approaches 2. Direct substitution results in a denominator of zero, indicating that we need to manipulate the expression. We start by multiplying the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{x} + \sqrt{2}\). This results in:

\[\frac{(x - 2)(\sqrt{x} + \sqrt{2})}{(\sqrt{x} - \sqrt{2})(\sqrt{x} + \sqrt{2})}\]

Multiplying the conjugates in the denominator gives us \(x - 2\). This allows us to cancel the common factor \(x - 2\) from the numerator and denominator, simplifying our expression to \(\sqrt{x} + \sqrt{2}\). We can now evaluate the limit by substituting \(x = 2\), yielding:

\[\sqrt{2} + \sqrt{2} = 2\sqrt{2}\]

Thus, the limit of the function as \(x\) approaches 2 is \(2\sqrt{2}\).

In another example, we find the limit of \(\frac{\sqrt{x + 9} - 3}{x}\) as \(x\) approaches 0. Direct substitution results in a zero denominator, prompting us to multiply by the conjugate of the numerator, \(\sqrt{x + 9} + 3\). This gives us:

\[\frac{(\sqrt{x + 9} - 3)(\sqrt{x + 9} + 3)}{x(\sqrt{x + 9} + 3)}\]

Multiplying the conjugates in the numerator results in \(x + 9 - 9 = x\). The expression simplifies to:

\[\frac{x}{x(\sqrt{x + 9} + 3)}\]

We can cancel the \(x\) terms, leading to:

\[\frac{1}{\sqrt{x + 9} + 3}\]

Now, substituting \(x = 0\) gives us:

\[\frac{1}{\sqrt{0 + 9} + 3} = \frac{1}{3 + 3} = \frac{1}{6}\]

Therefore, the limit of this rational function as \(x\) approaches 0 is \(\frac{1}{6}\).

In summary, when faced with rational functions that contain radicals and result in a zero denominator, the key steps involve multiplying by the conjugate, simplifying by canceling common factors, and then evaluating the limit. This method ensures that we can effectively find limits even in more complex scenarios.