Problem 44 - Calculus Early Transcendentals

Question

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{2 e^{x}}{e^{x}-5} $$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = \ln(5)\), Horizontal asymptote: \(y = 2\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. The denominator of the given function is \( e^x - 5 \). Set \( e^x - 5 = 0 \) to solve for \( x \): \( e^x = 5 \). Taking the natural logarithm of both sides gives \( x = \ln(5) \). Thus, the vertical asymptote is at \( x = \ln(5) \).

Step 2: Determine Horizontal Asymptotes

Horizontal asymptotes are established by inspecting the behavior of the function as \( x \) approaches infinity or negative infinity. As \( x \to \infty \), both \( e^x \) in the numerator and denominator grow exponentially, and the leading behavior is determined by the ratio of their coefficients, which is \( \frac{2}{1} = 2 \). Thus, the horizontal asymptote is at \( y = 2 \). As \( x \to -\infty \), the function approaches zero because \( e^x \) approaches zero in both the numerator and denominator.

Key Concepts

Vertical Asymptotes

To understand vertical asymptotes, envision a point on a graph where a curve shoots up to positive or negative infinity. This can happen where the denominator of a fraction approaches zero, creating a limitless climb or drop. For the given function, \[ y = \frac{2e^x}{e^x - 5} \] in a vertical asymptote emerges when the denominator equates to zero, while the numerator remains non-zero. By setting \[ e^x - 5 = 0, \] we solve to find \[ e^x = 5. \] Taking the natural logarithm of both sides, we derive \[ x = \ln(5). \] Hence, the vertical asymptote here is at the line \[ x = \ln(5). \] Think of this line as a barrier that the curve can never cross on the graph, as it fiercely shoots upward or downward just beside it.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a curve far out on the extremes of the x-axis, either heading towards positive or negative infinity. For the function \( y = \frac{2e^x}{e^x - 5} \), we investigate what happens when \( x \rightarrow \infty \) and \( x \rightarrow -\infty \). 💡 **Asymptotic Behavior:**- As \( x \rightarrow \infty, \) the term \( e^x \) in the numerator and denominator grows exceedingly large. In the limit, the behavior of the function depends on the ratio of the coefficient of \( e^x \), which is \( \frac{2}{1} = 2. \) Thus, the horizontal asymptote is located at \( y = 2. \)- In the case of \( x \rightarrow -\infty, e^x \) becomes negligible, leading the entire function towards zero. Hence, as \( x \rightarrow -\infty, y \rightarrow 0. \)Remember, horizontal asymptotes don't confine a curve as strictly as vertical ones; they merely suggest the direction the curve settles into over the long haul.

Exponential Functions

Exponential functions, characterized by expressions such as \( e^x, \)exhibit incredibly rapid growth or decay. Fundamental to many natural phenomena, they serve as building blocks for various mathematical concepts, including our function \( y = \frac{2e^x}{e^x - 5}. \)### Key characteristics of exponential functions include:- **Rapid Growth or Decay:** - As \( x \) increases, so does \( e^x, \) suggesting an accelerating upward trend. - Conversely, as \( x \) decreases, \( e^x \) swiftly drops towards zero. - **Continuous and Smooth Curves:** - Their graphs never break or abrupt, showcasing a smooth progression, whether increasing or decreasing.Exponential functions hold a special place in mathematics due to their unique growth patterns. With each step along the x-axis, they cover greater distances than their polynomial relatives. Understanding these traits can help you better comprehend their role in the overall structure of functions like the one we've just analyzed.