Question

Asymptotes Find the vertical and horizontal asymptotes of \(f(x)=e^{1 / x}\)

Short answer

Question: Determine the vertical and horizontal asymptotes of the function \(f(x) = e^{\frac{1}{x}}\). Answer: The function has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=1\).

Step-by-step solution

Identify the vertical asymptotes

To find the vertical asymptotes, we need to determine when the function goes to infinity. The function goes to infinity when the denominator of the exponent goes to zero. Thus, we need to find the values of \(x\) where the denominator is zero, i.e. \(\frac{1}{x}=0\). This occurs when \(x\) approaches zero. So, take the limit as \(x\) approaches to zero: $$ \lim_{x\to 0} e^{\frac{1}{x}} $$ Since as \(x\) approaches zero, \(\frac{1}{x}\) approaches infinity, the limit goes to infinity. Therefore, the function \(f(x)\) has a vertical asymptote at \(x=0\).

Identify the horizontal asymptotes

To find the horizontal asymptotes, we need to determine the limits of the function as \(x\) approaches positive and negative infinity: $$ \lim_{x\to +\infty} e^{\frac{1}{x}} \quad\text{and}\quad \lim_{x\to -\infty} e^{\frac{1}{x}} $$ As \(x\) approaches positive infinity, \(\frac{1}{x}\) approaches zero, and \(e^{\frac{1}{x}}\) approaches \(e^0 = 1\). Therefore, the function has a horizontal asymptote at \(y = 1\) as \(x\) goes to positive infinity. As \(x\) approaches negative infinity, \(\frac{1}{x}\) also approaches zero, and \(e^{\frac{1}{x}}\) approaches \(e^0 = 1\). Therefore, the function has a horizontal asymptote at \(y=1\) as \(x\) goes to negative infinity.

Write the final answer

In conclusion, the function \(f(x) = e^{\frac{1}{x}}\) has one vertical asymptote at \(x=0\) and one horizontal asymptote at \(y=1\).

Key concepts

Calculus

Calculus is a branch of mathematics focused on studying change and motion. It provides tools for understanding many different types of functions, like exponential functions, and analyzing their behaviors. Asymptotes are important in calculus because they help us understand a function's behavior as the input approaches certain critical points. These might be where the function becomes infinite or stabilizes to a certain value. Calculus involves two primary concepts: differentiation and integration. Differentiation helps find rates of change, which is useful for determining the slope of a function at any point. Integration, on the other hand, finds the total accumulated value, which can be helpful to determine areas under curves. In the context of asymptotes, we often use limits—a fundamental part of calculus—to analyze how a function behaves near particular points.

Limits

In calculus, limits are used to describe the behavior of functions as inputs approach specific values. They are essential for identifying asymptotes. Limits allow us to capture a function’s trend as the input grows very large, becomes very small, or approaches some critical value.When we talk about vertical asymptotes, such as where the function becomes undefined or infinite, limits can help identify these points. For example, in the exercise, when we see \(\lim_{x\to 0} e^{\frac{1}{x}}\), the limit goes to infinity as x approaches zero, indicating a vertical asymptote at x=0.Horizontal asymptotes are determined by examining the limits as the input (**x**) goes to positive or negative infinity. The exercise demonstrates that as \(x\to +\infty\) or \(x\to -\infty\), \(\frac{1}{x}\) approaches zero, making \(e^{\frac{1}{x}}\) approach \(e^0 = 1\). Hence, a horizontal asymptote at y=1 is found.

Exponential Functions

Exponential functions are a type of mathematical function characterized by a constant raised to the power of a variable. A classic example is the function \(f(x) = e^x\), where \(e\) is Euler's number, approximately 2.718.In the exercise, we're dealing with a slightly modified exponential function: \(f(x) = e^{\frac{1}{x}}\). This represents an exponential function where 1 divided by \(x\) serves as the exponent. Exponential functions have certain traits:

• They grow or decay at rates proportional to their size.
• They exhibit rapid changes, either growth or decay, around critical points like 0 in this exercise.
• They often involve asymptotic behavior, meaning they approach a line closely but never actually touch it.

Exponential functions are prevalent in modeling growth and decay processes in fields like biology, finance, and physics due to their dynamic behavior and versatility.