Question

c. does \\(\lim_{w \to 1} f(w)\\) exist? explain. choose the correct answer below. a. the limit does not exist because \\(\lim_{w \to 1^+} f(w) = \lim_{w \to 1^-} f(w)\\). b. the limit exists because \\(\lim_{w \to 1^+} f(w) = \lim_{w \to 1^-} f(w)\\). c. the limit exists because the postage for a letter that weighs 1 oz can be calculated exactly. d. the limit does not exist because \\(\lim_{w \to 1^+} f(w) \
eq \lim_{w \to 1^-} f(w)\\).

Explanation:

To determine if $\lim_{w\to1} f(w)$ exists, we check the left - hand limit ($\lim_{w\to1^{-}} f(w)$) and the right - hand limit ($\lim_{w\to1^{+}} f(w)$). The limit as $w$ approaches 1 exists if and only if $\lim_{w\to1^{+}} f(w)=\lim_{w\to1^{-}} f(w)$. Option B states that the limit exists because $\lim_{w\to1^{+}} f(w)=\lim_{w\to1^{-}} f(w)$, which is the correct condition for the existence of a limit. Option A says the limit does not exist when the one - sided limits are equal, which is wrong. Option C's reasoning about calculating postage exactly is not relevant to the limit definition. Option D says the limit does not exist because the one - sided limits are not equal, but the correct condition for existence is that the one - sided limits are equal.

Answer:

B. The limit exists because $\lim\limits_{w\to1^{+}} f(w)=\lim\limits_{w\to1^{-}} f(w)$