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The limit function you're asking about is:

\[
f(x) = \frac{8}{x} - 12
\]

We need to find the limit as \( x \to \infty \) and as \( x \to -\infty \).

### 1. Limit as \( x \to \infty \):
As \( x \) approaches positive infinity, the term \( \frac{8}{x} \) will approach 0. So, the limit of \( f(x) \) as \( x \to \infty \) becomes:

\[
\lim_{x \to \infty} \left( \frac{8}{x} - 12 \right) = 0 - 12 = -12
\]

### 2. Limit as \( x \to -\infty \):
As \( x \) approaches negative infinity, the term \( \frac{8}{x} \) will also approach 0. Thus, the limit of \( f(x) \) as \( x \to -\infty \) is:

\[
\lim_{x \to -\infty} \left( \frac{8}{x} - 12 \right) = 0 - 12 = -12
\]

So, for both limits, the result is \( -12 \).

Do you need further details or have additional questions on this? 

Here are 5 related questions:
1. What happens to the function when \( x \) approaches 0?
2. How would the limits change if the numerator were different, for example, \( 8x \)?
3. What is the behavior of rational functions as \( x \to \infty \) or \( x \to -\infty \)?
4. How do vertical asymptotes relate to limits in rational functions?
5. What is the difference between one-sided limits and two-sided limits?

Tip: Remember that when the denominator tends to infinity, the fraction generally tends to zero.

Find the limit of f(x) = 8/x - 12 as x approaches infinity and as x approaches negative infinity.

Explore how to find the limits of the function f(x) = 8/x - 12 as x approaches infinity and negative infinity. Understand the behavior of rational functions and their asymptotic properties. Suitable for high school students in Grades 10-12.

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