Determine all horizontal and vertical asymptotes. For each...

Determine all horizontal and vertical asymptotes. For each vertical asymptote, determine whether $f(x) \rightarrow \infty$ or $f(x) \rightarrow-\infty$ on either side of the asymptote.
$$f(x)=3 e^{-1 / x}$$

Key Concepts

Horizontal Asymptotes

A horizontal asymptote is a line y = k that the function approaches as x tends to infinity or negative infinity. In this context, evaluating the limit of the function as x ? ±? helps in determining the constant value that f(x) approaches, which is a key aspect of understanding end behavior in functions.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the function is undefined and the function values become unbounded (tending to ? or -?) as x approaches that value. Analyzing the limits from either side of these undefined points is essential to determine whether the function’s value diverges positively or negatively, or if differing behaviors occur on each side.

One-Sided Limit Analysis

One-sided limit analysis involves studying the behavior of a function as the input approaches a particular value from the left and from the right separately. This analysis is crucial when a function is not defined at a specific point, as it helps to identify potential discontinuities and the nature of any vertical asymptote.

Exponential Function Behavior

Understanding the properties of the exponential function is key when it comes to limits and asymptotic behavior. Exponential functions react predictably to large positive or negative exponents, which can force the function values to approach 0 or infinity. This behavior is instrumental in determining both horizontal and vertical asymptotes in functions involving exponentials.

Domain Considerations

Identifying the domain of a function is essential in asymptotic analysis, as points where the function is undefined can indicate locations of vertical asymptotes. Analyzing where and why the function fails to be defined (for example, due to division by zero) is a fundamental step in determining the asymptotic behavior of the function.

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