Question

Horizontal and vertical asymptotes. a. Analyze \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\) analyze \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{3 e^{x}+10}{e^{x}}$$

Short answer

The function $$f(x) = \frac{3e^x+10}{e^x}$$ has a horizontal asymptote at $$y = 3$$ and no vertical asymptotes.

Step-by-step solution

Find the horizontal asymptotes

To find the horizontal asymptotes, analyze the limits at infinity and minus infinity. $$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{3e^x + 10}{e^x}$$ $$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{3e^x + 10}{e^x}$$ Divide each term by $$e^x$$ in the numerator and the denominator. $$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{3 + \frac{10}{e^x}}{1}$$ $$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{3 + \frac{10}{e^x}}{1}$$ As $$x$$ approaches infinity, the term $$\frac{10}{e^x}$$ approaches 0. As $$x$$ approaches minus infinity, the term $$\frac{10}{e^x}$$ approaches infinity. So, we get: $$\lim_{x \to \infty} f(x) = 3$$ $$\lim_{x \to -\infty} f(x) = \infty$$ There is a horizontal asymptote at $$y = 3$$.

Find the vertical asymptotes

A vertical asymptote occurs when the denominator of the function is equal to zero. However, in this case, as the denominator is $$e^x$$, which is always positive, there is no vertical asymptote.

Analyze limits around each vertical asymptote (N/A)

Since there is no vertical asymptote, there are no limits around any vertical asymptote to analyze.

Summary

The function $$f(x) = \frac{3e^x+10}{e^x}$$ has a horizontal asymptote at $$y = 3$$ and no vertical asymptotes.

Key concepts

Horizontal Asymptotes

Horizontal asymptotes are a key feature of certain functions, indicating the value that a function approaches as the input, or x-value, goes to positive or negative infinity. In this case, we're looking at the function \[ f(x) = \frac{3e^x + 10}{e^x} \]To determine the horizontal asymptotes, we analyze the limits as \( x \to \infty \) and \( x \to -\infty \). When dividing each term by \( e^x \), the function simplifies to \( 3 + \frac{10}{e^x} \). As \( x \to \infty \), \( \frac{10}{e^x} \to 0 \), resulting in \( \lim_{x \to \infty} f(x) = 3 \). Thus, there is a horizontal asymptote at \( y = 3 \).
Horizontal asymptotes help to understand the long-term behavior of functions, especially when dealing with large numbers.

Vertical Asymptotes

Vertical asymptotes occur where a function's value grows without bound near a certain x-value, often meaning the denominator approaches zero. They represent values that the function never touches or crosses.
Considering the function \( f(x) = \frac{3e^x + 10}{e^x} \), the denominator \( e^x \) is never zero because exponential functions are always positive. Thus, for this specific function, there are no vertical asymptotes.
Identifying vertical asymptotes is crucial for avoiding undefined function values and understanding discontinuities in the function's graph.

Limits at Infinity

Limits at infinity give us insight into what values a function will approach as x approaches very large positive or negative numbers. This is key when identifying horizontal asymptotes or gauging behavior at the extremes.
For the function \( f(x) = \frac{3e^x + 10}{e^x} \), we consider two limits: \( \lim_{x \to \infty} \) and \( \lim_{x \to -\infty} \). As \( x \) tends towards infinity, the term \( \frac{10}{e^x} \) diminishes to zero, leading to the conclusion that \( \lim_{x \to \infty} f(x) = 3 \).
However, as \( x \to -\infty \), the entire function \( \frac{10}{e^x} \) increases indefinitely as \( e^x \to 0 \), causing \( \lim_{x \to -\infty} f(x) = \infty \). These calculations allow us to plot how a function behaves as x moves towards these extremes.

Exponential Functions

Exponential functions, like \( e^x \), are characterized by a constant base raised to a variable exponent. They have unique properties such as rapid growth or decay, depending on the sign of the exponent.
In the function \( f(x) = \frac{3e^x + 10}{e^x} \), \( e^x \) serves as both the numerator and denominator, simplifying the behavior of the function. Importantly, exponential functions never touch zero, meaning we avoid vertical asymptotes in many cases.
Understanding exponential functions is vital, as they frequently appear in scientific contexts like population growth, radioactive decay, and more, where these properties distinctly influence outcomes.

• Exponential growth occurs when the base exceeds 1.
• Exponential decay happens when the base is between 0 and 1.

Analyzing exponential functions helps predict trends and behaviors over time.