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) = \infty\\)
b. the limit does not exist and is neither \\(\infty\\) nor \\(-\infty\\).

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

Explanation:

Analyze the limit as \(x \to -3^-\)

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

Analyze the limit as \(x \to -3^+\)

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

Analyze the two-sided limit as \(x \to -3\)

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

Answer:

Question a

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

Question b

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

Question c

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