4. for the function g graphed in the accompanying figure, find (a) $limlimits_{x \to 0^-} g(x)$ (b) $limlimits_{x \to 0^+} g(x)$ (c) $limlimits_{x \to 0} g(x)$ (d) $g(0)$ (e) $limlimits_{x \to -\infty} g(x)$ (f) $limlimits_{x \to +\infty} g(x)$. figure ex-4

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### Part (a): $\boldsymbol{\lim_{x o 0^-} g(x)}$ # Explanation: ## Step1: Analyze left - hand limit To find the left - hand limit as $x$ approaches $0$ (i.e., $x o0^-$), we look at the behavior of the function $g(x)$ as $x$ gets closer to $0$ from the left side (values of $x$ less than $0$). From the graph, as $x$ approaches $0$ from the left, the function values approach the $y$ - value of the peak at $x = 0$ from the left. By observing the graph, we can see that the left - hand limit is the $y$ - coordinate of the point that the left - hand part of the graph approaches as $x$ approaches $0$. From the graph, when $x$ approaches $0$ from the left, $g(x)$ approaches $3$ (assuming the peak at $x = 0$ has a $y$ - value of $3$, we can infer this from the grid. If we consider the grid lines, the peak at $x = 0$ is at $y = 3$). $\lim_{x o 0^-} g(x)=3$ ### Part (b): $\boldsymbol{\lim_{x o 0^+} g(x)}$ # Explanation: ## Step1: Analyze right - hand limit To find the right - hand limit as $x$ approaches $0$ (i.e., $x o0^+$), we look at the behavior of the function $g(x)$ as $x$ gets closer to $0$ from the right side (values of $x$ greater than $0$). From the graph, as $x$ approaches $0$ from the right, the function values also approach the $y$ - value of the peak at $x = 0$. So, the right - hand limit is the same as the left - hand limit in terms of the $y$ - value it approaches. $\lim_{x o 0^+} g(x)=3$ ### Part (c): $\boldsymbol{\lim_{x o 0} g(x)}$ # Explanation: ## Step1: Recall the limit existence condition The limit $\lim_{x o a}f(x)$ exists if and only if $\lim_{x o a^-}f(x)=\lim_{x o a^+}f(x)$. We found that $\lim_{x o 0^-} g(x) = 3$ and $\lim_{x o 0^+} g(x)=3$. Since the left - hand limit and the right - hand limit are equal, the two - sided limit as $x$ approaches $0$ exists and is equal to this common value. $\lim_{x o 0} g(x)=3$ ### Part (d): $\boldsymbol{g(0)}$ # Explanation: ## Step1: Find the function value at $x = 0$ To find $g(0)$, we look at the $y$ - coordinate of the point on the graph of $y = g(x)$ when $x = 0$. From the graph, the point at $x = 0$ is the peak, and its $y$ - coordinate is $3$. So, $g(0)$ is the $y$ - value of the function at $x = 0$. $g(0)=3$ ### Part (e): $\boldsymbol{\lim_{x o -\infty} g(x)}$ # Explanation: ## Step1: Analyze the end - behavior as $x o-\infty$ To find the limit as $x$ approaches $-\infty$, we look at the behavior of the function as $x$ becomes very large in the negative direction. From the graph, as $x$ approaches $-\infty$, the left - most part of the graph approaches a horizontal asymptote or a specific $y$ - value. Looking at the left - hand side of the graph, as $x$ goes to $-\infty$, the function values approach $4$ (from the grid, we can see that the left - most horizontal part of the graph is at $y = 4$). $\lim_{x o -\infty} g(x)=4$ ### Part (f): $\boldsymbol{\lim_{x o +\infty} g(x)}$ # Explanation: ## Step1: Analyze the end - behavior as $x o+\infty$ To find the limit as $x$ approaches $+\infty$, we look at the behavior of the function as $x$ becomes very large in the positive direction. From the graph, as $x$ approaches $+\infty$, the right - most part of the graph is a line that is increasing. Wait, no, looking at the graph again, the right - most part of the graph (as $x o+\infty$) is a line that goes upwards? Wait, no, maybe I misread. Wait, the graph on the right - hand side, as $x$ increases beyond $5$, the line is going up. But wait, maybe the end - behavior: Wait, no, let's re - examine. Wait, the left - hand side has a horizontal part at $y = 4$, and the right - hand side, as $x o+\infty$, the function is increasing without bound? Wait, no, the graph shows that as $x$ approaches $+\infty$, the line is going up, so the limit as $x o+\infty$ of $g(x)$ is $+\infty$? Wait, no, maybe I made a mistake. Wait, the graph: the right - hand side, after $x = 5$, the line is going up, so as $x$ gets larger and larger (approaches $+\infty$), $g(x)$ goes to $+\infty$. But wait, maybe the graph is such that as $x o+\infty$, the function tends to $+\infty$. But let's check again. Wait, the left - hand side, as $x o-\infty$, the function approaches $4$ (the horizontal line on the left). The right - hand side, as $x o+\infty$, the line is increasing, so $\lim_{x o +\infty} g(x)=+\infty$? But maybe the graph is drawn with a line that has a positive slope as $x o+\infty$, so the function values keep increasing, so the limit is $+\infty$. But maybe I misinterpret the graph. Wait, the original graph: the left - most part is a horizontal line at $y = 4$, then it goes down, up, etc. The right - most part, after $x = 5$, is a line going up, so as $x o+\infty$, $g(x) o+\infty$. $\lim_{x o +\infty} g(x)=+\infty$ # Answers: (a) $\boldsymbol{3}$ (b) $\boldsymbol{3}$ (c) $\boldsymbol{3}$ (d) $\boldsymbol{3}$ (e) $\boldsymbol{4}$ (f) $\boldsymbol{+\infty}$

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Explanation:

Part (a): $\boldsymbol{\lim_{x \to 0^-} g(x)}$

Step1: Analyze left - hand limit

Part (b): $\boldsymbol{\lim_{x \to 0^+} g(x)}$

Step1: Analyze right - hand limit

Part (c): $\boldsymbol{\lim_{x \to 0} g(x)}$

Step1: Recall the limit existence condition

Part (d): $\boldsymbol{g(0)}$

Answer: