https://file.aiquickdraw.com/mathbot/file/1726168837811v3n2xuto.jpg

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.

Using the graph of the rational function y = f(x) given in the figure below, evaluate the limits. If you need to enter ∞ or -∞, type INFINITY or -INFINITY.

Analyze and evaluate the limits of a rational function using its graph. This guide covers key concepts such as horizontal and vertical asymptotes, and how to interpret the behavior of functions as they approach infinity or a specific value. Suitable for high school students in Grades 10-12.

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