The problem consists of two parts. Let's break down both parts:

Part (a):

You are asked to sketch a graph for a function $f(x)$ that satisfies several limits. Let's analyze them one by one:

1. $\lim_{x \to 1^-} f(x) = \infty$: As $x$ approaches 1 from the left, $f(x)$ tends towards $\infty$ (positive infinity).
2. $\lim_{x \to 1^+} f(x) = -\infty$: As $x$ approaches 1 from the right, $f(x)$ tends towards $-\infty$ (negative infinity).
3. $\lim_{x \to 2^-} f(x) = -\infty$: As $x$ approaches 2 from the left, $f(x)$ tends towards $-\infty$.
4. $\lim_{x \to 2^+} f(x) = -\infty$: As $x$ approaches 2 from the right, $f(x)$ also tends towards $-\infty$. This implies a vertical asymptote at $x = 2$ from both sides.
5. $\lim_{x \to 4} f(x) = \infty$: As $x$ approaches 4, $f(x)$ tends towards $\infty$.
6. $\lim_{x \to \infty} f(x) = -\infty$: As $x$ approaches infinity, $f(x)$ tends towards $-\infty$, indicating the behavior of the function far to the right.
7. $\lim_{x \to -\infty} f(x) = 2$: As $x$ approaches negative infinity, $f(x)$ approaches 2, suggesting a horizontal asymptote at $y = 2$.

Part (b):

You are asked to determine several limits based on your graph:

• $\lim_{x \to 1} f(x)$
• $\lim_{x \to 2} f(x)$
• $\lim_{x \to 4^-} f(x)$
• $\lim_{x \to 4^+} f(x)$

These can be interpreted from the graph after sketching the behavior of the function according to the given limits.

Would you like me to generate a visual sketch of the graph based on these conditions?