To determine whether there's a limit in the expression \(\frac{x - 2}{x - 4}\), we need to consider the behavior of the function as \(x\) approaches various values.

### 1. **Limit as \(x \to 4\):**
   - Substituting \(x = 4\) directly into the expression gives:
     \[
     \frac{4 - 2}{4 - 4} = \frac{2}{0}
     \]
     This results in division by zero, which means the expression is undefined at \(x = 4\).
   - However, the limit as \(x\) approaches 4 from the left (\(x \to 4^-\)) and from the right (\(x \to 4^+\)) behave differently:
     - As \(x \to 4^-\), the expression tends towards \(-\infty\).
     - As \(x \to 4^+\), the expression tends towards \(+\infty\).
   - **Conclusion**: The limit as \(x \to 4\) does not exist because the left-hand and right-hand limits are not equal.

### 2. **Limit as \(x \to \infty\):**
   - If \(x\) increases without bound (i.e., \(x \to \infty\)), the expression becomes:
     \[
     \frac{x - 2}{x - 4} \approx \frac{x}{x} = 1
     \]
   - **Conclusion**: The limit as \(x \to \infty\) is \(1\).

### 3. **Limit as \(x \to -\infty\):**
   - If \(x\) decreases without bound (i.e., \(x \to -\infty\)), the expression similarly becomes:
     \[
     \frac{x - 2}{x - 4} \approx \frac{x}{x} = 1
     \]
   - **Conclusion**: The limit as \(x \to -\infty\) is also \(1\).

### Summary:
- The limit as \(x \to 4\) does not exist because the function approaches different infinities from the left and right.
- The limit as \(x \to \infty\) and \(x \to -\infty\) is \(1\).

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

1. What are the different types of limits in calculus?
2. How can you determine if a function is continuous at a point?
3. What are one-sided limits and how are they used?
4. How do limits at infinity differ from limits at a finite value?
5. What role do vertical asymptotes play in determining limits?

**Tip:** When evaluating limits, it's important to consider the direction from which the variable approaches the value (left or right) and whether the function tends towards a specific value or infinity.

Explore the behavior of the expression \(\frac{x - 2}{x - 4}\) through detailed limit analysis. Discover why the limit as \(x\) approaches 4 does not exist due to different infinities from left and right approaches, while the limits as \(x\) approaches infinity are \(1\). Ideal for advanced high school and college students.

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