1. The problem asks to sketch a function $g(x)$ with specific limit behaviors and discontinuities. 2. Let's analyze each condition and describe the graph behavior: - As $x \to -\infty$, $g(x) \to +\infty$. This means the left end of the graph goes upward without bound. - At $x = 0$, $\lim_{x \to 0} g(x) = 0$ from the left, and $g(-1) = 0$. So near $x=0$ from the left side, the function approaches 0, and the point at $x=-1$ is exactly 0. - At $x=2$, there is a jump discontinuity with $\lim_{x \to 2^-} g(x) = -2$ and $\lim_{x \to 2^+} g(x) = 2$. The function jumps from -2 to 2 at $x=2$. - At $x=-1$, $\lim_{x \to -1} g(x) = -1$ but $g(1)$ is undefined (DNE). This means the limit exists at $x=-1$ but the function is not defined at $x=1$. - At $x=0$, $g(x)$ is continuous with $g(0) = 1$ and $\lim_{x \to 0} g(x) = 1$. This means the function value and limit match at $x=0$. - There is a removable discontinuity at $x=-4$ with $g'( -4 ) = -2$. This means the function has a hole at $x=-4$ but the derivative exists and equals -2. - At $x=1/2$, there is a jump discontinuity and $g(1/2) = 3$. The function jumps at $x=1/2$ and the value is 3. - At $x=-3$, there is an infinite discontinuity and $g(-3) = -1$. The function tends to infinity near $x=-3$ but is defined as -1 at that point. - At $x=5$, the function is continuous and $\lim_{x \to 5} g(x) = -5$. The function value at 5 is -5 and the limit matches. 3. To sketch this function, start from the left: - Draw the graph going up to infinity as $x \to -\infty$. - Include an infinite discontinuity at $x=-3$ with a vertical asymptote and a defined point at $g(-3)=-1$. - Include a removable discontinuity (hole) at $x=-4$ with slope $-2$. - At $x=-1$, the limit is $-1$ and $g(-1)=0$ (from earlier condition), so the function approaches -1 but the value at -1 is 0. - At $x=0$, the function is continuous with value and limit 1. - At $x=1/2$, a jump discontinuity with $g(1/2)=3$. - At $x=2$, another jump discontinuity from -2 to 2. - Finally, at $x=5$, the function is continuous with value and limit -5. This description guides the sketch of $g(x)$ with the specified behaviors and discontinuities.