https://file.aiquickdraw.com/mathbot/file/17302025131626apmlqgu.jpg

The graph appears to be of a rational function, likely something like \( f(x) = \frac{1}{x - 2} \) or a similar function with a vertical asymptote at \( x = 2 \).

For such a function, here are the limits as \( x \) approaches \( 2 \):

1. **Right-hand limit**: 
   \[
   \lim_{x \to 2^+} f(x) = +\infty
   \]
   This indicates that as \( x \) approaches \( 2 \) from the right, \( f(x) \) increases without bound.

2. **Left-hand limit**: 
   \[
   \lim_{x \to 2^-} f(x) = -\infty
   \]
   This indicates that as \( x \) approaches \( 2 \) from the left, \( f(x) \) decreases without bound.

3. **Two-sided limit**: 
   Since the left-hand and right-hand limits are not equal (one goes to \( +\infty \) and the other to \( -\infty \)), the two-sided limit does not exist:
   \[
   \lim_{x \to 2} f(x) \text{ does not exist}
   \]

These statements characterize the behavior of the function around \( x = 2 \) based on the observed asymptote.

Would you like further details on understanding these limits or applying them to similar functions? 

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### Additional Questions
1. What is the definition of a right-hand and left-hand limit?
2. Why do functions with vertical asymptotes often have limits that approach infinity?
3. How can you determine if a function has a vertical asymptote at a specific point?
4. What is the significance of two-sided limits and when can they be used?
5. How does the concept of a limit apply to real-world scenarios?

### Tip
When examining limits involving infinity, focus on whether the function grows positively or negatively as it approaches the vertical asymptote. This distinction clarifies the behavior near points where the function is undefined.

Explore limit statements as x approaches 2 for a rational function with a vertical asymptote at x = 2. This problem covers limits from the left and right sides, examining behavior as the function approaches infinity or negative infinity. Suitable for students in Grades 11-12.

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