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 ($\lim_{w\to1^+} f(w)$). A limit exists at a point if and only if the left - hand limit and the right - hand limit are equal, i.e., $\lim_{w\to a} f(w)$ exists $\iff \lim_{w\to a^-} f(w)=\lim_{w\to a^+} f(w)$.

• Option A: It says the limit does not exist because $\lim_{w\to1^+} f(w)=\lim_{w\to1^-} f(w)$. But if the left and right limits are equal, the limit should exist. So A is wrong.
• Option B: It states that the limit exists because $\lim_{w\to1^+} f(w)=\lim_{w\to1^-} f(w)$. This is the correct condition for the existence of a limit.
• Option C: The ability to calculate the postage exactly has nothing to do with the existence of the limit of the function $f(w)$ as $w\to1$. So C is wrong.
• Option D: It says the limit does not exist because $\lim_{w\to1^+} f(w)

eq\lim_{w\to1^-} f(w)$, but this is the opposite of the correct condition for non - existence (if they are not equal, the limit does not exist, but the option's reasoning is not relevant to the limit definition here and also the statement about the limit not existing due to unequal one - sided limits is misused in the context of the option's explanation).

Answer:

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