Question

use the graph below to determine whether the statements about the function y = f(x) are true or false.
true or false: \\( \lim\limits_{x \to -3^+} f(x) = 9 \\).
\\( \circ \\) true (checked) \\( \circ \\) false
true or false: \\( \lim\limits_{x \to 0^-} f(x) = 3 \\).
\\( \circ \\) false (checked) \\( \circ \\) true
true or false: \\( \lim\limits_{x \to 0^-} f(x) = \lim\limits_{x \to 0^+} f(x) \\).
\\( \circ \\) false \\( \circ \\) true (checked)
true or false: \\( \lim\limits_{x \to 0} f(x) \\) exists.
\\( \circ \\) true \\( \circ \\) false

Explanation:

To determine the truth of the statement \(\lim_{x \to 0} f(x)\) exists, we analyze the left - hand limit (\(\lim_{x \to 0^{-}} f(x)\)) and the right - hand limit (\(\lim_{x \to 0^{+}} f(x)\)):

Step 1: Recall the definition of the existence of a limit

For a limit \(\lim_{x \to a} f(x)\) to exist, the left - hand limit \(\lim_{x \to a^{-}} f(x)\) and the right - hand limit \(\lim_{x \to a^{+}} f(x)\) must exist and be equal, i.e., \(\lim_{x \to a^{-}} f(x)=\lim_{x \to a^{+}} f(x)\).

Step 2: Analyze the left - hand limit as \(x\to0^{-}\)

From the graph, as \(x\) approaches \(0\) from the left (values of \(x\) less than \(0\) and getting closer to \(0\)), we observe the behavior of the function \(y = f(x)\). The function is a parabola - like curve approaching \(0\) (the \(y\) - value approaches \(0\)) as \(x\to0^{-}\).

Step 3: Analyze the right - hand limit as \(x\to0^{+}\)

As \(x\) approaches \(0\) from the right (values of \(x\) greater than \(0\) and getting closer to \(0\)), the function (the same parabola - like curve) also approaches \(0\) (the \(y\) - value approaches \(0\)) as \(x\to0^{+}\).

Step 4: Compare the left - hand and right - hand limits

Since \(\lim_{x \to 0^{-}} f(x) = 0\) and \(\lim_{x \to 0^{+}} f(x)=0\), we have \(\lim_{x \to 0^{-}} f(x)=\lim_{x \to 0^{+}} f(x)\). By the definition of the existence of a limit, when the left - hand limit and the right - hand limit at a point are equal, the limit at that point exists.

Answer:

True