Question

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

Explanation:

First Statement: $\boldsymbol{\lim_{x \to -3^+} f(x) = 9}$

Step1: Understand Right-Hand Limit

The right-hand limit as $x \to -3^+$ means we look at the values of $f(x)$ as $x$ approaches $-3$ from values greater than $-3$ (i.e., moving towards $-3$ from the right side on the x - axis).

Step2: Analyze the Graph at $x = -3$ (Right - Hand Side)

From the graph, when we approach $x=-3$ from the right (values like $x=-2.9, - 2.99$, etc.), the function's graph is approaching the point with $y$-value $9$. So the right - hand limit $\lim_{x \to -3^+} f(x)$ is indeed $9$.

Second Statement: $\boldsymbol{\lim_{x \to 0^-} f(x) = 3}$

Step1: Understand Left-Hand Limit

The left - hand limit as $x \to 0^-$ means we look at the values of $f(x)$ as $x$ approaches $0$ from values less than $0$ (i.e., moving towards $0$ from the left side on the x - axis).

Step2: Analyze the Graph at $x = 0$ (Left - Hand Side)

From the graph, when we approach $x = 0$ from the left (values like $x=-0.1,-0.01$, etc.), the function's graph is approaching the vertex of the parabola - like shape at $x = 0$. The $y$-value at that point (the limit as $x$ approaches $0$ from the left) is $0$ (since the vertex is at $(0,0)$), not $3$. So the statement $\lim_{x \to 0^-} f(x)=3$ is false.

Answer:

s:

• For $\lim_{x \to -3^+} f(x) = 9$: True
• For $\lim_{x \to 0^-} f(x) = 3$: False