Question

the graph of f shown below has vertical asymptotes at \\(x = -3\\) and \\(x = 5\\). analyze the following limits.

a. \\(\lim_{x \to -3^-} f(x)\\)
b. \\(\lim_{x \to -3^+} f(x)\\)
c. \\(\lim_{x \to -3} f(x)\\)
d. \\(\lim_{x \to 5^-} f(x)\\)
e. \\(\lim_{x \to 5^+} f(x)\\)
f. \\(\lim_{x \to 5} f(x)\\)

a. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \\(\lim_{x \to -3^-} f(x) = \infty\\)
b. the limit does not exist and is neither \\(\infty\\) nor \\(-\infty\\).

b. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \\(\lim_{x \to -3^+} f(x) = \square\\)
b. the limit does not exist and is neither \\(\infty\\) nor \\(-\infty\\).

Explanation:

Analyze the behavior of the function as \(x\) approaches \(-3\) from the right

$$ \lim_{x \to -3^+} f(x) $$

Looking at the graph, there is a vertical asymptote at \(x = -3\). As \(x\) approaches \(-3\) from the right (values of \(x\) slightly greater than \(-3\)), the curve goes downwards towards negative infinity.

Determine the limit value

$$ \lim_{x \to -3^+} f(x) = -\infty $$

Match with the given multiple-choice options

The options for part b are:

• A. \(\lim_{x \to -3^+} f(x) = \text{box}\)
• B. The limit does not exist and is neither \(\infty\) nor \(-\infty\).

Since the limit is \(-\infty\), option A is the correct choice, and the value to fill in the box is \(-\infty\).

Answer:

• (A) \(\lim_{x \to -3^+} f(x) = -\infty\) (Correct answer)
• (B) The limit does not exist and is neither \(\infty\) nor \(-\infty\).