Limits at infinity refer to the behavior of a function as the input values approach positive or negative infinity. This concept helps to understand how functions behave when they grow very large or very small, which is crucial for analyzing end behavior, asymptotes, and continuity. It enables mathematicians to determine the horizontal asymptotes of a function and predict its long-term trends.
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Limits at infinity can indicate whether a function approaches a specific finite value or diverges to infinity.
If a rational function has a degree of the numerator less than the degree of the denominator, its limit at infinity is 0.
Conversely, if the degree of the numerator is greater than the degree of the denominator, the limit at infinity is either positive or negative infinity.
For functions with equal degrees in the numerator and denominator, the limit at infinity is determined by the leading coefficients.
Graphically, limits at infinity can be analyzed by examining how a graph behaves far to the left and right on the coordinate plane.
Review Questions
How do you determine if a rational function has a limit at infinity, and what does this imply about its end behavior?
To determine if a rational function has a limit at infinity, compare the degrees of the numerator and denominator. If the numerator's degree is less than that of the denominator, the limit approaches 0, indicating that the function flattens out. If it's greater, the limit approaches positive or negative infinity, suggesting that the function grows without bound. If they are equal, then you take the ratio of leading coefficients for your limit.
Describe how horizontal asymptotes relate to limits at infinity and provide an example illustrating this relationship.
Horizontal asymptotes indicate where a function approaches as its input values go to positive or negative infinity. For instance, in the function $$f(x) = \frac{2x^2 + 3}{4x^2 + 5}$$, both numerator and denominator have equal degrees, so as $$x$$ approaches infinity, $$f(x)$$ approaches $$\frac{2}{4} = \frac{1}{2}$$. Therefore, there’s a horizontal asymptote at $$y = \frac{1}{2}$$.
Evaluate how understanding limits at infinity can aid in graphing functions and predicting their long-term behavior.
Understanding limits at infinity is essential for accurately graphing functions because it allows you to identify horizontal asymptotes and anticipate how the function behaves far away from the origin. By knowing whether a function approaches specific values or diverges, you can sketch more accurate graphs. For example, if you know that as $$x$$ approaches infinity a function approaches a certain y-value, you can ensure your graph reflects this behavior as it stretches outward in either direction.
A horizontal line that a graph approaches as the input values go to infinity or negative infinity, indicating the value that the function approaches.
Vertical Asymptote: A vertical line that a function approaches as the output values approach infinity or negative infinity, often occurring at points where the function is undefined.