Limits at Infinity
End behaviour of functions.
Key idea
End behaviour of functions.
In depth
As x grows large, compare the dominant terms. For rational functions, divide by the highest power of x in the denominator. The limit gives the horizontal asymptote.
Worked example
lim (3x² + 1)/(5x² − 2) = 3/5, dividing top and bottom by x².
Formula
Real-world uses
- Predicting long-run population, drug concentration or average cost as the size grows
Common mistake
Treating ∞ as a number. ∞ − ∞ and ∞/∞ are indeterminate and need work.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Limits at Infinity?
End behaviour of functions. It is a topic at the advanced level in Calculus.
What is the formula for Limits at Infinity?
The key formula is lim_{x→∞} 1/x = 0.
Where is Limits at Infinity used in real life?
Real-world examples: Predicting long-run population, drug concentration or average cost as the size grows.
What should I learn before Limits at Infinity?
Make sure you are comfortable with Factoring and Rationalizing Limits, the previous topic in Calculus.
What is a common mistake with Limits at Infinity?
Treating ∞ as a number. ∞ − ∞ and ∞/∞ are indeterminate and need work.
How can I practise Limits at Infinity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Limits at Infinity?
Move on to Infinite Limits and Asymptotes next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.
- Squeeze Theorem – If g ≤ f ≤ h and g, h → L then f → L.