Find the horizontal asymptotes of each function using limits at infinity.
f(x) = (3e^5x + 7e^6x) / (9e^5x + 14e^6x)

Verified step by step guidance

1. Identify the highest power of the exponential function in both the numerator and the denominator. In this case, it's \( e^{6x} \).
2. Factor \( e^{6x} \) out of both the numerator and the denominator.
3. Rewrite the function as \( f(x) = \frac{e^{6x}(\frac{3}{e^x} + 7)}{e^{6x}(\frac{9}{e^x} + 14)} \).
4. Cancel \( e^{6x} \) from the numerator and the denominator, simplifying the expression to \( f(x) = \frac{\frac{3}{e^x} + 7}{\frac{9}{e^x} + 14} \).
5. Evaluate the limit of \( f(x) \) as \( x \to \infty \). As \( x \to \infty \), \( \frac{3}{e^x} \to 0 \) and \( \frac{9}{e^x} \to 0 \), so the limit becomes \( \frac{0 + 7}{0 + 14} = \frac{7}{14} = \frac{1}{2} \). Thus, the horizontal asymptote is \( y = \frac{1}{2} \).

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input approaches infinity or negative infinity. They indicate the value that the function approaches, providing insight into its long-term behavior. To find horizontal asymptotes, one typically evaluates the limit of the function as x approaches infinity.

Limits at Infinity

Limits at infinity involve determining the value that a function approaches as the variable grows indefinitely large or small. This concept is crucial for analyzing the end behavior of functions, especially rational functions, where the degrees of the numerator and denominator can dictate the limit's outcome.

Exponential Functions

Exponential functions, such as e^(kx), grow or decay at rates proportional to their current value. In the context of limits, the behavior of these functions as x approaches infinity is significant, as they can dominate polynomial terms, influencing the overall limit and thus the horizontal asymptote of the function.