Graph each function and then find the specified limits. When...

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=\frac{1}{x}-2 ; \text { find } \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow 0} f(x).$$

Key Concepts

Limits of Functions

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing the continuity of functions and their behavior near points of interest, including at infinity or near discontinuities.

Limit at Infinity

The limit at infinity describes how a function behaves as the input grows without bound. In many cases, this behavior results in the function approaching a horizontal asymptote, which is critical for understanding long?term trends and end behavior in rational and other types of functions.

Limit at a Point and Discontinuities

Investigating the limit of a function as the input approaches a specific finite value, especially when the function is undefined at that point, highlights the behavior around discontinuities. This analysis is important for determining whether a limit exists and for identifying vertical asymptotes or other forms of discontinuity.

Graphing Functions and Asymptotic Behavior

Graphing functions, particularly those with rational expressions, requires understanding how transformations (such as translations and reflections) affect the basic shape of the graph. Recognizing vertical and horizontal asymptotes from these graphs helps in visualizing the function's behavior near discontinuities and at the extremes.

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