Use limits to find the equations of the Horizontal Asymptotes h(x) = (4(2^x) + 16)/6-2^x

To find the horizontal asymptotes of the function \( h(x) = \frac{4(2^x) + 16}{6 - 2^x} \), we need to evaluate the behavior of \( h(x) \) as \( x \to \infty \) and \( x \to -\infty \).

### Step 1: Find the limit as \( x \to \infty \)
As \( x \to \infty \), the term \( 2^x \) grows exponentially, so we focus on the highest power of \( 2^x \) in both the numerator and the denominator.

\[
\lim_{x \to \infty} h(x) = \lim_{x \to \infty} \frac{4(2^x) + 16}{6 - 2^x}
\]

In the numerator, \( 4(2^x) \) dominates, and in the denominator, \( -2^x \) dominates as \( 2^x \) becomes very large. So, we approximate the function:

\[
\lim_{x \to \infty} h(x) \approx \lim_{x \to \infty} \frac{4(2^x)}{-2^x} = \lim_{x \to \infty} \frac{4}{-1} = -4
\]

Thus, as \( x \to \infty \), the horizontal asymptote is:

\[
y = -4
\]

### Step 2: Find the limit as \( x \to -\infty \)
As \( x \to -\infty \), \( 2^x \to 0 \) because exponential functions decay to zero for negative large values of \( x \).

\[
\lim_{x \to -\infty} h(x) = \lim_{x \to -\infty} \frac{4(2^x) + 16}{6 - 2^x}
\]

Since \( 2^x \to 0 \), the function simplifies to:

\[
\lim_{x \to -\infty} h(x) = \frac{4(0) + 16}{6 - 0} = \frac{16}{6} = \frac{8}{3}
\]

Thus, as \( x \to -\infty \), the horizontal asymptote is:

\[
y = \frac{8}{3}
\]

### Conclusion:
The horizontal asymptotes of the function \( h(x) \) are:
- \( y = -4 \) as \( x \to \infty \)
- \( y = \frac{8}{3} \) as \( x \to -\infty \)

Would you like further clarification or more details on this?

Here are 5 related questions to deepen your understanding:
1. How do vertical asymptotes differ from horizontal asymptotes?
2. What role does the exponential term play in determining horizontal asymptotes?
3. Can you find the oblique asymptote of a function?
4. How does the behavior of exponential functions differ as \( x \to \infty \) and \( x \to -\infty \)?
5. How can limits be used to determine the asymptotes of rational functions?

**Tip:** Always compare the growth rates of the terms in the numerator and denominator to identify asymptotic behavior efficiently!

Learn how to use limits to find the horizontal asymptotes of the exponential function h(x) = (4(2^x) + 16)/(6 - 2^x). This solution covers limits, exponential growth, and asymptotic behavior for advanced high school students in Grades 10-12.

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