Limit at Infinity
In short: A limit at infinity reports the height a graph levels off at when x runs endlessly to the right or to the left, and so it pinpoints any horizontal asymptote.
AP Calculus ABCalculusAP Calculus AB Unit 1
What is Limit at Infinity?
A limit at infinity tells you the long-run behavior of a function: what height the graph levels off at far out to the right, or far out to the left. Infinity is not a number you can plug in; it is a direction in which \(x\) keeps growing.
We write \(\lim_{x \to \infty} f(x) = L\) when the outputs stay as close to \(L\) as desired once \(x\) is large enough. The line \(y = L\) is then a horizontal asymptote. If the outputs grow without bound instead, the result is written \(\infty\) or \(-\infty\), and no finite value exists.
How to spot it
- The question asks for a limit at infinity: what happens to f(x) when x becomes extremely large or extremely negative.
- A rational function has the same highest power in the numerator and denominator.
- A story asks about long-term behavior, such as the amount of medicine in the blood after many hours.
- A graph flattens toward a dashed horizontal line on the far right or far left.
The method
- Identify the highest power of \(x\) in the denominator.
- Divide each term, top and bottom, by that power, as a limit at infinity of a fraction requires.
- Replace each fraction of the form \(\frac{c}{x^k}\) by 0, since it vanishes for huge \(x\).
- Read off the remaining ratio; if the numerator’s degree is larger, the outputs grow without bound.
Limit at Infinity: solved example
Problem
Find \(\lim_{x \to \infty}\dfrac{6x^2 – 5x}{2x^2 + 9}\).
Solution
The highest power in the denominator is \(x^2\). Divide every term by it: \(\dfrac{6 – \frac{5}{x}}{2 + \frac{9}{x^2}}\).
As \(x\) grows, \(\frac{5}{x} \to 0\) and \(\frac{9}{x^2} \to 0\), leaving \(\frac{6}{2}\). The value is 3, so \(y = 3\) is a horizontal asymptote.
Try it
Question 1
Find \(\lim_{x \to -\infty}\frac{8}{x – 3}\).
Show the answer
The denominator grows large in size, so the fraction shrinks to 0.
Question 2
Find \(\lim_{x \to \infty}\frac{x^3 + 1}{5x^2}\).
Show the answer
The numerator’s degree is higher, so the outputs grow without bound: \(\infty\).
Limit at Infinity vs. Asymptote
The line is a picture on the graph, while the limit is a number computed with algebra. Evaluating the long-run height gives the number L, and then \(y = L\) is the asymptote. A graph can approach the line from above or below, and for some functions it even crosses the line before settling.
Keep going
Lessons
Worksheets and quizzes
- Infinite Limits and Asymptotes – Worksheet
- Infinite Limits and Asymptotes – Quiz
- Evaluating Limits Algebraically – Worksheet
- Evaluating Limits Algebraically – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus