Problem 66 - Calculus: Early Transcendentals

Question

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(x)=e^{1 / x}$$

Step-by-Step Solution

Answer

Answer: The vertical asymptote of the function $$g(x) = e^{\frac{1}{x}}$$ is at x = 0.

Step 1: Identify the function

The function is given as $$g(x) = e^{\frac{1}{x}}$$. It is an exponential function with base e.

Step 2: Find the domain of the function

The domain of the function is the set of all possible values of x for which the function is defined. The function will be undefined when the exponent (i.e., the denominator of the fractional exponent) is equal to 0, which is when x = 0. Therefore, the domain of the function is all real numbers except x = 0, which can be written as $$x \neq 0$$.

Step 3: Examine the behavior of the function as x approaches the value where it is undefined

We will now analyze the behavior of the function as x approaches the value where it is undefined. This is important because vertical asymptotes occur when the function approaches infinity as x approaches the undefined value. Let's examine the function as x approaches 0 from the left and the right. As x approaches 0 from the right (x -> 0+), the exponent $$\frac{1}{x}$$ approaches infinity, since the denominator (x) is becoming very small and positive. Therefore, the function approaches: $$g(x) = e^{\infty} \to \infty$$ As x approaches 0 from the left (x -> 0-), the exponent $$\frac{1}{x}$$ approaches negative infinity, since the denominator (x) is becoming very small and negative. In this situation, the function approaches: $$g(x) = e^{-\infty} \to 0$$

Step 4: Identify the vertical asymptote(s)

From the analysis in step 3, we can see that the function approaches infinity as x approaches 0 from the right. Therefore, there is a vertical asymptote at x = 0.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In the function \( g(x) = e^{\frac{1}{x}} \), the base \( e \) is known as Euler's number, approximately equal to 2.718. This makes \( g(x) \) an exponential function of the form \( e^x \), but here, the exponent itself is a fraction \( \frac{1}{x} \).

Exponential functions have characteristic growth behavior. For example, if the exponent is positive and increasing, the function grows rapidly. Conversely, if the exponent is negative and decreasing, the function approaches zero. This is quite different from polynomial functions, where the rate of increase or decrease depends on the degree of the polynomial.

Understanding exponential functions can help predict the behavior of variables in natural phenomena, such as population growth or radioactive decay. Familiarizing oneself with structure and properties of exponential functions is crucial in tackling problems related to vertical asymptotes.

Domain of a Function

The domain of a function is the complete set of possible values of the input, usually represented by \( x \), for which the function is defined. For the function \( g(x) = e^{\frac{1}{x}} \), identifying the domain involves understanding for which \( x \) values the function doesn't have undefined expressions or mathematical anomalies.

In this case, the fraction \( \frac{1}{x} \) becomes undefined when \( x = 0 \). Hence, the function does not exist at this point because division by zero is not possible. As a result, the domain of \( g(x) \) includes all real numbers except \( 0 \).

To express this, we write: \( x eq 0 \). Recognizing the domain helps understand where the function operates normally and identifies points that may require special attention due to undefined values, critical for analyzing vertical asymptotes.

Limits

Limits help us understand the behavior of a function as it approaches a particular point. In the function \( g(x) = e^{\frac{1}{x}} \), we use limits to examine what happens as \( x \) approaches the value where the function is undefined, which is \( x = 0 \).

When taking the limit of \( g(x) \) as \( x \to 0^{+} \) (approaching from the right), \( \frac{1}{x} \) becomes a very large positive number, causing \( e^{\frac{1}{x}} \) to grow indefinitely towards infinity.

Conversely, as \( x \to 0^{-} \) (approaching from the left), \( \frac{1}{x} \) becomes a very large negative number, which makes \( e^{\frac{1}{x}} \) shrink towards zero. This dichotomy indicates a dramatic change in the value of the function.

Understanding these limits is vital for identifying vertical asymptotes, as they describe how a function behaves near points where it isn’t fully defined. This knowledge helps predict where the graph of a function shoots up or down indefinitely.

Undefined Values

In many mathematical functions, certain input values can create anomalies or undefined expressions. These undefined values arise when mathematical operations cannot yield a result, typically because they require impossible calculations, such as division by zero.

In the case of \( g(x) = e^{\frac{1}{x}} \), the expression becomes undefined when \( x = 0 \). At this point, the exponent cannot be calculated because dividing by zero is not allowed. These undefined values are pivotal in understanding the behavior of a function around these points, often indicating vertical asymptotes.

Breaking down the behavior of functions near these undefined values helps mathematicians and students anticipate and correct for infinite growth or shrinkage, which is a common feature in vertical asymptotes. Recognizing these situations enables a more comprehensive analysis of how functions behave around critical points on their graphs.