https://file.aiquickdraw.com/mathbot/file/1738518659728hh0vgnhp.jpg

I see that you've uploaded a graph of a function, and you're asking for help determining various limits based on it. Let me analyze the graph and solve the limits one by one.

Here are the limits you asked about:

### a. \( \lim_{{x \to 2}} f(x) \)
From the graph, check the value of the function as \( x \) approaches 2. The function appears to approach a value of 1 as \( x \to 2 \). Hence, the limit is:
\[
\lim_{{x \to 2}} f(x) = 1
\]

### b. \( \lim_{{x \to 3^+}} f(x) \)
For this limit, we need to examine the value of \( f(x) \) as \( x \) approaches 3 from the right. The graph shows that as \( x \) approaches 3 from the right, the value of \( f(x) \) approaches \( +\infty \). So, we have:
\[
\lim_{{x \to 3^+}} f(x) = +\infty
\]

### c. \( \lim_{{x \to 3^-}} f(x) \)
Here, we need to check the value of \( f(x) \) as \( x \) approaches 3 from the left. The graph shows that as \( x \) approaches 3 from the left, \( f(x) \) approaches 1. Therefore:
\[
\lim_{{x \to 3^-}} f(x) = 1
\]

### d. \( \lim_{{x \to 3}} f(x) \)
Since the left-hand limit and the right-hand limit at \( x = 3 \) are not equal (the right-hand limit is \( +\infty \), while the left-hand limit is 1), the two-sided limit does not exist. Hence:
\[
\lim_{{x \to 3}} f(x) = \text{DNE (Does Not Exist)}
\]

### e. \( \lim_{{x \to 0}} f(x) \)
At \( x = 0 \), the graph shows the function approaches \( +\infty \). Therefore:
\[
\lim_{{x \to 0}} f(x) = +\infty
\]

### f. \( \lim_{{x \to 0^+}} f(x) \)
Looking at the graph from the right of 0, the function approaches \( +\infty \). Thus:
\[
\lim_{{x \to 0^+}} f(x) = +\infty
\]

### g. \( \lim_{{x \to 0^-}} f(x) \)
From the graph, as \( x \) approaches 0 from the left, \( f(x) \) tends towards \( -\infty \). So:
\[
\lim_{{x \to 0^-}} f(x) = -\infty
\]

### h. \( \lim_{{x \to \infty}} f(x) \)
As \( x \to \infty \), the graph shows that \( f(x) \) approaches 1. Therefore:
\[
\lim_{{x \to \infty}} f(x) = 1
\]

### i. \( \lim_{{x \to -\infty}} f(x) \)
As \( x \to -\infty \), the graph shows the function approaches 2. Hence:
\[
\lim_{{x \to -\infty}} f(x) = 2
\]

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Would you like further details on any of these steps or more explanations on limits? Feel free to ask!

### Related Questions:
1. What does it mean for a limit to not exist (DNE)?
2. How do you handle one-sided limits when the two-sided limit does not exist?
3. How do limits at infinity differ from finite limits?
4. What are the conditions under which a function has an infinite limit?
5. What do vertical asymptotes represent in terms of limits?

**Tip:** Always remember to check both the left-hand and right-hand limits separately when analyzing a limit at a specific point.

For the function f whose graph is given, determine the following limits: a. lim (x→2) f(x), b. lim (x→3⁺) f(x), c. lim (x→3⁻) f(x), d. lim (x→3) f(x), e. lim (x→0) f(x), f. lim (x→0⁺) f(x), g. lim (x→0⁻) f(x), h. lim (x→∞) f(x), i. lim (x→-∞) f(x).

Explore how to determine limits from a given graph of a function. This detailed solution covers key mathematical concepts such as one-sided limits, infinite limits, and asymptotic behavior. Suitable for high school students in Grades 10-12.

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