Let's go through each part of the question by analyzing the graph shown in the image.

The graph shows a rational function with different behaviors at different parts of the graph. We will evaluate the limits accordingly:

(a) $\lim_{x \to \infty} f(x)$

As $x$ approaches $\infty$ (positive infinity), the function approaches a horizontal asymptote near the $x$-axis. This indicates that:

$\lim_{x \to \infty} f(x) = 0$

(b) $\lim_{x \to -\infty} f(x)$

As $x$ approaches $-\infty$ (negative infinity), the function again approaches a horizontal asymptote near the $x$-axis. Therefore:

$\lim_{x \to -\infty} f(x) = 0$

(c) $\lim_{x \to 7^-} f(x)$

As $x$ approaches 7 from the left ($x \to 7^-$), the function drops down towards negative infinity. This indicates that:

$\lim_{x \to 7^-} f(x) = -\infty$

(d) $\lim_{x \to 7^+} f(x)$

As $x$ approaches 7 from the right ($x \to 7^+$), the function shoots upwards towards positive infinity. Hence:

$\lim_{x \to 7^+} f(x) = \infty$

Summary of Answers:

(a) $\lim_{x \to \infty} f(x) = 0$
(b) $\lim_{x \to -\infty} f(x) = 0$
(c) $\lim_{x \to 7^-} f(x) = -\infty$
(d) $\lim_{x \to 7^+} f(x) = \infty$

Would you like any further clarifications or additional details?

Here are 5 related questions to expand your understanding:

1. How do you determine horizontal asymptotes from a graph of a rational function?
2. What is the significance of limits as $x$ approaches positive and negative infinity?
3. How can you find vertical asymptotes algebraically from the equation of a rational function?
4. What does it mean for a function to approach $\infty$ or $-\infty$ as $x$ nears a certain value?
5. How does the behavior of a function change near its asymptotes?

Tip: Remember that vertical asymptotes indicate where a function approaches infinity or negative infinity, while horizontal asymptotes help determine the function's end behavior as $x$ tends to infinity.