Question

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

Explanation:

1. Statement: $\boldsymbol{\lim_{x\to 0^-} f(x) = \lim_{x\to 0^+} f(x)}$

To check the left - hand limit ($\lim_{x\to 0^-} f(x)$) and the right - hand limit ($\lim_{x\to 0^+} f(x)$), we look at the graph of $y = f(x)$ as $x$ approaches 0 from the left (values less than 0) and from the right (values greater than 0). From the graph, as $x$ approaches 0 from both the left and the right, the function approaches the same $y$ - value. So, $\lim_{x\to 0^-} f(x)=\lim_{x\to 0^+} f(x)$.

The limit of a function $\lim_{x\to a} f(x)$ exists if and only if $\lim_{x\to a^-} f(x)=\lim_{x\to a^+} f(x)$. From the previous statement, we know that $\lim_{x\to 0^-} f(x)=\lim_{x\to 0^+} f(x)$. So, by the definition of the existence of a limit, $\lim_{x\to 0} f(x)$ exists.

From the graph, when $x$ approaches 0 (both from the left and the right), the function $y = f(x)$ approaches a $y$ - value of 0 (the vertex of the parabola - like part of the graph is at $(0,0)$). So, $\lim_{x\to 0} f(x) = 0$.

Answer:

True

2. Statement: $\boldsymbol{\lim_{x\to 0} f(x)}$ exists