Limits at Infinity
Limits at infinity describe end behavior and horizontal asymptotes.
About this topic
A limit at infinity describes the end behavior of a function as x grows without bound. If lim x→∞ f(x) = L, the line y = L is a horizontal asymptote. For rational functions, compare degrees: if the denominator's degree is larger the limit is 0, if equal it is the ratio of leading coefficients, and if the numerator's degree is larger the limit is infinite.
A reliable method is to divide every term by the highest power of x in the denominator and then let 1/x terms go to zero. For (3x² + 5)/(2x² − x), dividing by x² gives (3 + 5/x²)/(2 − 1/x), which approaches 3/2. Exponentials dominate polynomials, which in turn dominate logarithms, so eˣ/x² grows without bound while ln x / x tends to 0.
Real-world uses include predicting the long-term population of a species with limited resources, the terminal velocity of a falling object, or the steady-state temperature of cooling coffee.
Tips: a function can cross its horizontal asymptote in the middle of the graph, and the limit as x goes to negative infinity may differ from the limit as x goes to positive infinity.
Key ideas
- Compare degrees of numerator and denominator
- 1/x → 0 as x → ∞
- Divide by the highest power
Quick check
Use cases: where Limits at Infinity helps
- Homework: review the key idea behind Limits at Infinity before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Limits at Infinity without notes.
- Visual learning: play with the interactive lab to see how Limits at Infinity changes when you adjust its inputs.
- Teaching a friend: explain Limits at Infinity in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Limits at Infinity?
Limits at infinity describe end behavior and horizontal asymptotes.
What will I learn in Limits at Infinity?
Compare degrees of numerator and denominator. 1/x → 0 as x → ∞. Divide by the highest power.
Is Limits at Infinity part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Limits at Infinity lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Limits at Infinity?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.