Question

true or false: $\lim\limits_{x \to 0^-} f(x) = \lim\limits_{x \to 0^+} f(x)$
$\circ$ false
$\checkmark$ true

true or false: $\lim\limits_{x \to 0} f(x)$ exists.
$\checkmark$ true
$\circ$ false

true or false: $\lim\limits_{x \to 0} f(x) = 0$
$\circ$ true
$\circ$ false

Explanation:

For the question "True or false: $\lim\limits_{x\to 0} f(x) = 0$"

Step1: Recall limit definition

For a limit $\lim\limits_{x\to a} f(x)$ to equal $L$, the left - hand limit $\lim\limits_{x\to a^{-}} f(x)$ and the right - hand limit $\lim\limits_{x\to a^{+}} f(x)$ must both exist and be equal to $L$. From the graph (the parabola - shaped part near $x = 0$), we can see the behavior of the function as $x$ approaches $0$ from the left and the right. The vertex of the parabola (the lowest point of the parabola - shaped graph) at $x = 0$ has a $y$ - value (the value of the function) that is not $0$. Let's assume the parabola is of the form $y=x^{2}$ (just for the shape analysis, since it's a parabola opening upwards with vertex at $(0,0)$? Wait, no, looking at the graph grid, the vertex seems to be at $(0,0)$? Wait, no, the $y$ - axis on the graph: the vertical axis has markings. Wait, maybe I misread. Wait, the first two questions: $\lim\limits_{x\to 0^{-}}f(x)=\lim\limits_{x\to 0^{+}}f(x)$ (True) and $\lim\limits_{x\to 0}f(x)$ exists (True). Now, for $\lim\limits_{x\to 0}f(x)$, what is its value? From the graph, the function near $x = 0$ (the parabola) has its vertex at $x = 0$. If the parabola is, say, $y = x^{2}$, then at $x = 0$, $y = 0$? Wait, no, maybe the graph's vertex is at $(0,0)$. Wait, but let's think again. Wait, the first two questions: the left - hand limit and right - hand limit as $x\to0$ are equal (so the limit exists). Now, what is the value of the limit? If the function near $x = 0$ (the parabolic part) has its minimum at $x = 0$, and from the graph, the $y$ - coordinate of the vertex (the point where $x = 0$ on the parabola) is $0$? Wait, no, maybe I made a mistake. Wait, the third question: "True or false: $\lim\limits_{x\to 0}f(x)=0$". Let's analyze the graph. The graph on the left is a parabola opening upwards with vertex at $(0,0)$ (since the vertex is at $x = 0$, and the lowest point is at $y = 0$). So as $x$ approaches $0$ from the left ($x\to0^{-}$) and from the right ($x\to0^{+}$), the function values approach $0$. So the limit as $x\to0$ of $f(x)$ is $0$. Wait, but maybe the graph is different. Wait, the first two questions: the first one is "True" (left and right limits equal), the second is "True" (limit exists). Now, for the third one, if the vertex of the parabola is at $(0,0)$, then $\lim\limits_{x\to 0}f(x)=0$, so the answer is True? Wait, no, maybe I misread the graph. Wait, the user's graph: the left - hand side is a parabola, and at $x = 0$, the vertex is at $(0,0)$? Let's check the grid. The $x$ - axis is from - 10 to 10, and the $y$ - axis: the parabola has a vertex at $x = 0$, and the $y$ - value at the vertex: looking at the grid, the vertical lines (y - axis) have markings. If the parabola is, for example, $y=x^{2}$, then at $x = 0$, $y = 0$. So as $x$ approaches $0$ from both sides, the function approaches $0$. So the limit is $0$, so the answer is True? Wait, but maybe the graph is different. Wait, the first two questions: the first is True (left and right limits equal), the second is True (limit exists). Now, the third question: "True or false: $\lim\limits_{x\to 0}f(x)=0$". So if the left - hand limit and right - hand limit are equal (from first question) and the limit exists (second question), and the value of the limit (the common value of left and right limits) is $0$ (from the graph of the parabola with vertex at $(0,0)$), then the answer is True.

Answer:

True (the option with "True" for the question "True or false: $\lim\limits_{x\to 0} f(x) = 0$")