To find the limit of the expression \(\frac{x - 2}{x - 4}\) as \(x\) approaches a certain value, we first need to specify the value of \(x\) that you want to approach. Typically, limits are evaluated as \(x\) approaches a specific number, such as \(x \to 4\), \(x \to \infty\), or another value. 

Please specify the value of \(x\) you want to approach. If you meant to find the limit as \(x\) approaches a particular value (e.g., \(x \to 4\)), let me know!

If \(x \to 4\) is what you intended, here's how you would proceed:

1. Substitute \(x = 4\) into the expression:
   \[
   \frac{4 - 2}{4 - 4} = \frac{2}{0}
   \]
   This results in an undefined expression since division by zero is not allowed.

   When an expression results in something like \(\frac{2}{0}\), it typically indicates that the limit may not exist, or it could suggest the presence of a vertical asymptote.

2. For the limit as \(x\) approaches 4:
   - If you approach 4 from the left (\(x \to 4^-\)), the expression tends towards negative infinity (\(-\infty\)).
   - If you approach 4 from the right (\(x \to 4^+\)), the expression tends towards positive infinity (\(+\infty\)).

   Therefore, the limit does not exist at \(x = 4\).

Would you like more details or have any questions about this limit? Here are some related questions:

1. How do you evaluate limits that result in indeterminate forms like \(\frac{0}{0}\)?
2. What is the significance of one-sided limits in calculus?
3. How do you determine the existence of a vertical asymptote?
4. What happens if you approach infinity in a limit problem?
5. How can you apply L'Hôpital's Rule to limits?

**Tip:** Always check if direct substitution leads to an indeterminate form (like \(\frac{0}{0}\)) or an undefined expression (like \(\frac{a}{0}\)) before evaluating the limit.

Explore how to find the limit of the expression (x - 2) / (x - 4) as x approaches a specific value. Learn about indeterminate forms, vertical asymptotes, and the behavior of limits approaching a point. Ideal for advanced high school students and calculus enthusiasts.

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