It looks like you're working on sketching a function that satisfies a set of limit conditions and discontinuities. I'll guide you step by step on how to construct this graph.

Key Features to Include in the Graph:

1. Asymptotes

   • Draw a dashed vertical asymptote at $x = -6$, since $\lim\limits_{x \to -6} f(x) = \infty$.
   • Draw a dashed horizontal asymptote at $y = 1$, since $\lim\limits_{x \to -\infty} f(x) = 1$.

2. Discontinuities and Limits

   • Jump discontinuity at $x = -1$

     • $\lim\limits_{x \to -1^-} f(x) = 3$, meaning as $x$ approaches $-1$ from the left, $f(x)$ is close to $3$.
     • $\lim\limits_{x \to -1^+} f(x) = -2$, meaning as $x$ approaches $-1$ from the right, $f(x)$ is close to $-2$.
     • $f(-1) = 0$, so we plot the point $(-1, 0)$.

   • Infinite discontinuity at $x = 3$

     • $\lim\limits_{x \to 3^-} f(x) = 2$ and $\lim\limits_{x \to 3^+} f(x)$ does not exist (which likely means it heads toward $\pm \infty$).

   • Removable discontinuity at $x = 7$

     • $\lim\limits_{x \to 7^-} f(x) = 1$, but $f(7) = 4$, so there is a hole at $(7,1)$ and a point at $(7,4)$.

3. Other Key Points and Behavior

   • $f(1) = 0$ → The function crosses the x-axis at $(1,0)$.
   • $\lim\limits_{x \to \infty} f(x) = \infty$, meaning the function increases indefinitely as $x \to \infty$.

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How the Graph Looks:

• The leftmost part of the function approaches the horizontal asymptote $y = 1$.
• A jump discontinuity at $x = -1$ from $y=3$ to $y=-2$.
• The function moves through $(-1, 0)$ and $(1,0)$, with an increasing trend.
• At $x = 3$, there is an infinite discontinuity, meaning a vertical asymptote.
• At $x = 7$, there is a removable discontinuity, with a hole at $(7,1)$ and a separate point at $(7,4)$.
• As $x \to \infty$, the function grows toward infinity.

Would you like me to draw this graph for you?