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.