The limit at infinity, denoted as $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$, refers to the behavior of a function as the input variable approaches positive or negative infinity. It describes the value that the function approaches as the input variable gets arbitrarily large or small.
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The limit at infinity is used to determine the long-term behavior of a function and can be used to identify horizontal asymptotes.
If $\lim_{x \to \infty} f(x) = L$, then the function has a horizontal asymptote at $y = L$.
Limits at infinity can be evaluated using algebraic techniques, such as factoring, simplifying, or using L'Hôpital's rule.
Limits at infinity can be used to analyze the growth or decay of functions, such as exponential and rational functions.
The limit at infinity may not exist if the function oscillates or does not approach a specific value as the input variable approaches positive or negative infinity.
Review Questions
Explain the significance of the limit at infinity in the context of 12.1 Finding Limits: Numerical and Graphical Approaches.
The limit at infinity is a crucial concept in the context of 12.1 Finding Limits: Numerical and Graphical Approaches. It allows us to analyze the long-term behavior of functions and determine the existence of horizontal asymptotes. By understanding the limit at infinity, we can use both numerical and graphical approaches to investigate the limiting behavior of functions as the input variable approaches positive or negative infinity. This information can provide valuable insights into the overall shape and properties of the function, which is essential for understanding the function's behavior and making accurate predictions.
Describe how the limit at infinity can be used to identify horizontal asymptotes.
The limit at infinity is directly related to the concept of horizontal asymptotes. If the limit of a function as the input variable approaches positive or negative infinity is a finite value, L, then the function has a horizontal asymptote at $y = L$. This means that the graph of the function will approach the horizontal line $y = L$ as the input variable gets arbitrarily large or small. Understanding the relationship between the limit at infinity and horizontal asymptotes is crucial in 12.1 Finding Limits: Numerical and Graphical Approaches, as it allows us to analyze the long-term behavior of functions and make accurate predictions about their properties.
Analyze how the evaluation of limits at infinity can be used to study the growth or decay of functions, such as exponential and rational functions, in the context of 12.1 Finding Limits: Numerical and Graphical Approaches.
In the context of 12.1 Finding Limits: Numerical and Graphical Approaches, the evaluation of limits at infinity can provide valuable insights into the growth or decay of functions, such as exponential and rational functions. By analyzing the limit of a function as the input variable approaches positive or negative infinity, we can determine the long-term behavior of the function and its rate of growth or decay. For example, the limit of an exponential function at positive infinity will approach positive infinity, indicating exponential growth, while the limit at negative infinity will approach zero, indicating exponential decay. Similarly, the limit of a rational function at positive or negative infinity can reveal the asymptotic behavior of the function and its suitability for modeling certain real-world phenomena. Understanding these properties through the evaluation of limits at infinity is essential for interpreting the behavior of functions in the context of 12.1 Finding Limits: Numerical and Graphical Approaches.