Question

hw4 the limit of a function from tables and graphs (target score: 2/5 answered: 2/5 question 3 the graph below is the function f(x) find \\( \lim_{x \to 2^-} f(x) \\) find \\( \lim_{x \to 2^+} f(x) \\) find \\( \lim_{x \to 2} f(x) \\) find \\( f(2) \\) question help: video message instructor

Explanation:

For $\boldsymbol{\lim_{x \to 2^-} f(x)}$:

Step1: Analyze left - hand limit

To find the left - hand limit as $x$ approaches $2$ (denoted as $x\to2^-$), we look at the values of the function $f(x)$ as $x$ gets closer to $2$ from the left side (values of $x$ less than $2$). From the graph, as $x$ approaches $2$ from the left, the function approaches the $y$ - value of the open circle at $x = 2$ from the left. The $y$ - coordinate of the open circle (when approaching from the left) is $- 2$.
<No formula needed, just graphical analysis>

Step2: Conclusion for left - hand limit

So, $\lim_{x\to2^-}f(x)=-2$.

For $\boldsymbol{\lim_{x \to 2^+} f(x)}$:

Step1: Analyze right - hand limit

To find the right - hand limit as $x$ approaches $2$ (denoted as $x\to2^+$), we look at the values of the function $f(x)$ as $x$ gets closer to $2$ from the right side (values of $x$ greater than $2$). From the graph, as $x$ approaches $2$ from the right, the function approaches the $y$ - value of the open circle at $x = 2$ from the right. The $y$ - coordinate of the open circle (when approaching from the right) is $- 2$.
<No formula needed, just graphical analysis>

Step2: Conclusion for right - hand limit

So, $\lim_{x\to2^+}f(x)=-2$.

For $\boldsymbol{\lim_{x \to 2} f(x)}$:

Step1: Recall the limit existence condition

The limit of a function $\lim_{x\to a}f(x)$ exists if and only if $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$. We already found that $\lim_{x\to2^-}f(x)=-2$ and $\lim_{x\to2^+}f(x)=-2$.

Step2: Determine the limit

Since the left - hand limit and the right - hand limit are equal, $\lim_{x\to2}f(x)=-2$.

For $\boldsymbol{f(2)}$:

Answer:

s:

• $\lim_{x\to2^-}f(x)=\boldsymbol{-2}$
• $\lim_{x\to2^+}f(x)=\boldsymbol{-2}$
• $\lim_{x\to2}f(x)=\boldsymbol{-2}$
• $f(2)=\boldsymbol{-4}$