For the function f whose graph is given, state the following

(a)
lim
x → ∞
f(x)
(b)
lim
x → −∞
f(x)
(c)
lim
x → 1
f(x)
(d)
lim
x → 3
f(x)
(e) the equations of the asymptotes (Enter your answers as a comma-separated list of equations.)
vertical:
horizontal:

why do you bother posting questions about a graph we cannot see?

what a waste of time.

To determine the limits and equations of asymptotes for the given function, you need to analyze the behavior of the graph near the given values. Here's how you can find the answers:

(a) To find the limit as x approaches positive infinity (x → ∞), observe the behavior of the graph on the right-hand side. If the graph goes towards a specific value as x becomes large, that would be the limit. If the graph does not approach any specific value and continues to increase or decrease without bound, the limit does not exist.

(b) To find the limit as x approaches negative infinity (x → -∞), observe the behavior of the graph on the left-hand side. Similar to the previous case, if the graph approaches a specific value as x becomes more negative, that would be the limit. If the graph does not approach any specific value and continues to increase or decrease without bound, the limit does not exist.

(c) To find the limit as x approaches 1 (x → 1), check the behavior of the graph as x gets closer and closer to 1 from both the left and right sides. If the graph approaches a specific value from both sides, that would be the limit. If the graph approaches different values from the left and right sides, or if it does not approach any value at all, the limit does not exist.

(d) To find the limit as x approaches 3 (x → 3), follow the same approach as in part (c). Examine the behavior of the graph as x gets closer and closer to 3 from both sides. If the graph approaches a specific value from both sides, that would be the limit. If the graph approaches different values from the left and right sides, or if it does not approach any value at all, the limit does not exist.

(e) To determine the equations of the asymptotes, there are two possibilities: vertical and horizontal asymptotes.

- Vertical asymptotes occur when the function approaches infinity or negative infinity (increases or decreases without bound) as x approaches a specific value. To find the equation of a vertical asymptote, identify the x-value where the function exhibits this behavior. The equation will be x = a, where a is the x-value.

- Horizontal asymptotes occur when the function approaches a specific value as x approaches positive or negative infinity. To find the equation of a horizontal asymptote, observe the behavior of the graph as x becomes very large or very negative. If the graph approaches a specific value as x → ∞ or x → -∞, that value will be the equation of the horizontal asymptote. You can express this as y = b, where b is the value.

To determine the equations of the asymptotes for the given function, you would need to analyze the behavior of the graph and identify any vertical and horizontal asymptotes present. Unfortunately, without the actual graph of the function, it is not possible to provide the specific equations of the asymptotes.

To determine the limits of the function f at certain values and the equations of the asymptotes, we need to analyze the behavior of the graph.

(a) To find the limit as x approaches positive infinity, look at the end behavior of the graph. In this case, the graph approaches the value 2. Therefore,

lim(x → ∞) f(x) = 2.

(b) To find the limit as x approaches negative infinity, again observe the end behavior. The graph approaches the value -1. Hence,

lim(x → -∞) f(x) = -1.

(c) To find the limit as x approaches 1, determine the value of f(x) when x is very close to 1. The graph appears to approach the value 3. Consequently,

lim(x → 1) f(x) = 3.

(d) Similarly, to find the limit as x approaches 3, observe the behavior of the graph. It seems that the graph approaches positive infinity. Therefore,

lim(x → 3) f(x) = ∞.

(e) Next, let's determine the equations of the asymptotes.

Vertical asymptotes occur when the function approaches positive or negative infinity as x approaches a certain value. From the graph, it is clear that there are no vertical asymptotes.

Horizontal asymptotes happen when the function approaches a specific value as x approaches positive or negative infinity. In this case, the graph approaches the horizontal line y = 2. Therefore,

Vertical asymptotes: None
Horizontal asymptotes: y = 2.