Question

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

Short answer

Answer: The vertical asymptote is at \(x = 0\) and the horizontal asymptote is at \(y = 1\).

Step-by-step solution

Find vertical asymptote

We will analyze the behavior of \(f(x)\) as \(x\) approaches \(0\). In this case, the exponent in the function \(f(x) = e^{1/x}\) becomes very large, since dividing by a small number (\(x \approx 0\)) results in a large value for the exponent. Let's consider the limits: 1. \(\lim_{x \to 0^+} e^{1/x}\) 2. \(\lim_{x \to 0^-} e^{1/x}\) For the first limit, as \(x \to 0^+\), \(1/x \to +\infty\), and the limit becomes: \(\lim_{x \to 0^+} e^{1/x} = e^{+\infty} = +\infty\). For the second limit, as \(x \to 0^-\), \(1/x \to -\infty\), and the limit becomes: \(\lim_{x \to 0^-} e^{1/x} = e^{-\infty} = 0\). Since the limit does not exist as \(x \to 0\), there is a vertical asymptote at \(x = 0\).

Find horizontal asymptotes

Now, we will analyze the behavior of \(f(x)\) as \(x\) approaches \(\pm\infty\). In this case, the exponent in the function \(f(x) = e^{1/x}\) becomes very small, since dividing by a large number (\(x \approx \pm\infty\)) results in a small value for the exponent. Let's consider the limits: 1. \(\lim_{x \to +\infty} e^{1/x}\) 2. \(\lim_{x \to -\infty} e^{1/x}\) For the first limit, as \(x \to +\infty\), \(1/x \to 0^{+}\), and the limit becomes: \(\lim_{x \to +\infty} e^{1/x} = e^{0} = 1\). For the second limit, as \(x \to -\infty\), \(1/x \to 0^{-}\), and the limit becomes: \(\lim_{x \to -\infty} e^{1/x} = e^{0} = 1\). Since the limits exist and are equal, there is a horizontal asymptote at \(y = 1\).

Conclusion

The function \(f(x) = e^{1/x}\) has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 1\).

Key concepts

Exponential Functions

Exponential functions are mathematical expressions written in the form of \(f(x) = e^{g(x)}\), where \(e\) is a constant approximately equal to 2.71828. Exponential functions are known for their growth or decay rate, which can be rapid depending on the exponent. In our example, \(f(x) = e^{1/x}\), the exponent is \(1/x\). This makes the behavior of the function unique, as the value of the exponent changes dramatically as \(x\) approaches zero or infinity.
Exponential functions often appear in situations involving continuous growth or decay, such as populations, investments, or radioactive decay. Understanding the nature of the exponent is key to dissecting these functions. The function \(e^{1/x}\) exhibits interesting behavior due to the inverse relationship of \(1/x\), dramatically affecting the curve based on different \(x\) values.

Limits and Continuity

Limits help us understand what happens to a function as the input value approaches a particular point. For exponential functions like \(f(x) = e^{1/x}\), determining limits is crucial to identify asymptotes. As \(x\) approaches zero or infinity, analyzing limits helps us comprehend the behavior of the function.
To determine the function's limits, we substitute values close to the target \(x\) value. This is particularly important in understanding discontinuities, such as vertical asymptotes.

• If the limit results in a finite value as \(x\) approaches infinity, it indicates a horizontal asymptote.
• If the limit does not exist or becomes infinitely large or small as \(x\) approaches a particular value, it may suggest a vertical asymptote.

Limits guide us in sketching the overall shape of functions and can highlight where discontinuities may occur.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) moves towards positive or negative infinity. They are horizontal lines that the graph of the function gets closer to but never quite reaches as \(x\) becomes very large or very small. For \(f(x) = e^{1/x}\), calculating the limits as \(x\) approaches infinity tells us if and where horizontal asymptotes exist.
In this example, as \(x\) tends towards positive or negative infinity:

• The exponent \(1/x\) approaches zero, thus the function \(e^{1/x}\) approaches \(e^0 = 1\).

This means there is a horizontal asymptote at \(y = 1\). Understanding horizontal asymptotes allows us to predict long-term behavior of functions, a crucial insight in many fields such as physics and economics.

Vertical Asymptotes

Vertical asymptotes occur where the function tends towards infinity as \(x\) approaches a particular value. These are vertical lines the function cannot cross. To find them for the function \(f(x) = e^{1/x}\), we look at the behavior as \(x\) approaches zero.
In this scenario:

• As \(x\) approaches zero from the positive side, \(1/x\) becomes very large, causing \(e^{1/x}\) to shoot up towards positive infinity.
• Conversely, as \(x\) approaches zero from the negative side, \(1/x\) becomes very negative, leading \(e^{1/x}\) towards zero.

Since the limits from both sides don't align, it confirms a vertical asymptote at \(x = 0\). Vertical asymptotes are crucial in understanding the natural constraints of functions and in predicting anomalies in real-world models.