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: $\lim_{{x \to -\infty}} f(x) = 2$

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.