Limit at infinity

Definition

A limit at infinity refers to the behavior of a function as the input approaches positive or negative infinity. It helps determine the end behavior of functions, showing what value the function gets closer to as the input becomes larger or smaller without bound. Understanding limits at infinity is crucial for analyzing the continuity and overall behavior of functions, particularly rational functions and their asymptotes.

5 Must Know Facts For Your Next Test

1. Limits at infinity can reveal horizontal asymptotes for rational functions, indicating the value a function approaches as x goes to positive or negative infinity.
2. If a function approaches a finite limit as x approaches infinity, it suggests that the graph will level off at that value.
3. Rational functions can have different limits at infinity based on their degrees; if the degree of the numerator is less than the degree of the denominator, the limit will be 0.
4. When both the numerator and denominator of a rational function have the same degree, the limit at infinity is determined by the leading coefficients.
5. Understanding limits at infinity helps in sketching graphs and understanding long-term trends in data represented by functions.

Review Questions

• How do limits at infinity help in understanding the end behavior of rational functions?
  • Limits at infinity provide insights into how rational functions behave as their inputs become very large or very small. By analyzing these limits, we can determine horizontal asymptotes, which indicate the values the function approaches at extremes. For example, if the degree of the numerator is less than that of the denominator, the limit is 0, suggesting that the function flattens out towards the x-axis.
• Discuss how to find horizontal asymptotes using limits at infinity for different types of rational functions.
  • To find horizontal asymptotes using limits at infinity, evaluate the limit of a rational function as x approaches positive or negative infinity. If the degree of the numerator is less than that of the denominator, the limit is 0, indicating a horizontal asymptote at y = 0. If both degrees are equal, divide their leading coefficients to find the asymptote. If the numerator's degree exceeds that of the denominator, there is no horizontal asymptote.
• Evaluate and analyze a given rational function's limit at infinity and discuss its implications on continuity and graph behavior.
  • To evaluate a rational function's limit at infinity, substitute large values for x and observe its behavior. For example, consider f(x) = (2x^2 + 3)/(x^2 + 5). As x approaches infinity, both terms grow similarly, leading to a limit of 2 (the ratio of leading coefficients). This result indicates a horizontal asymptote at y = 2, suggesting that as x grows larger, f(x) will approach this line. This informs us about continuity in that there are no breaks or jumps as x heads toward infinity, and enhances our understanding of how to sketch or interpret graphs accurately.

Related terms

Asymptote: A line that a graph approaches as it heads towards infinity. There are vertical, horizontal, and oblique asymptotes that describe the behavior of functions at extremes.

Continuity: A property of a function that means it does not have any breaks, jumps, or holes in its graph, ensuring it can be drawn without lifting a pencil.

Infinite Limits: A concept that describes the behavior of a function approaching infinity or negative infinity as the input approaches a specific point.