Limits (AA HL)
A limit describes the value a function or sequence settles towards as its input grows without bound (or approaches a particular point). At AA HL this mostly means evaluating limits at infinity, where the trick is spotting which terms matter and which vanish. This page focuses on that skill, with worked examples and the mistakes that lose the most marks. It's part of the broader Series & Limits topic.
19 questions on this sub-topic.
Two ways to handle a limit
Covered under IB syllabus references AHL5.12 (understanding of limits, convergence and divergence) and AHL5.13 (evaluation of limits of the form \(\lim_{x\to a}\frac{f(x)}{g(x)}\), including the indeterminate form \(\infty/\infty\) that a rational function gives at infinity). Neither technique below is a memorised formula from the booklet - both are methods you apply to the expression in front of you.
Rational functions at infinity
Divide every term by the highest power of \(x\) present, then let \(x\to\infty\).
Any term of the form \(\tfrac{1}{x^n}\) tends to \(0\), leaving a ratio of the remaining coefficients.
Standard exponential limit
\(\displaystyle\lim_{n\to\infty}\left(1+\dfrac{a}{n}\right)^{n}=e^{a}\)
A standard result, not in the formula booklet - recognise the pattern and read off \(a\).
Want the full topic overview and GDC guidance for this area? See Series & Limits.
Worked examples
Evaluate \(\displaystyle\lim_{x\to\infty}\dfrac{3x^2 + 2x}{5x^2 - 1}.\)
Worked solution
Divide by \(x^2\): M1
\(\lim\dfrac{3 + 2/x}{5 - 1/x^2}\) A1
as \(x\to\infty\) the \(1/x\) terms vanish, A1
\(= \dfrac{3}{5}.\) A1
Find \(\displaystyle\lim_{n\to\infty}\left(1 + \dfrac{3}{n}\right)^{n}.\)
Worked solution
Using \(\lim_{n\to\infty}(1 + \tfrac{a}{n})^n = e^{a}\) with \(a\) M1
\(= 3\): A1
the limit is \(e^{3}\) A1
\(\approx 20.1.\) A1
Common mistakes
- Dividing only the numerator by the highest power. Every term on both sides of the fraction must be divided, not just the top - otherwise the ratio changes.
- Assuming a \(\tfrac{\infty}{\infty}\) form is automatically \(1\). The result depends entirely on the degrees and leading coefficients of the numerator and denominator - it could be \(0\), a finite ratio, or diverge.
- Misreading the value of \(a\) in \((1+a/n)^n\). A sign error or missing coefficient here changes the final answer from \(e^{a}\) to a completely different power of \(e\).
- Assuming every limit exists. If the highest power in the numerator is greater than in the denominator, the limit is \(\pm\infty\), not a finite number - check the degrees of both before assuming a finite answer exists at all - only when the degrees of the numerator and denominator match does the limit settle on the ratio of their leading coefficients, rather than diverging to infinity or collapsing to zero.
Ready to practise properly?
20 limits questions, marked instantly like the real exam.
Quick answers
How do you evaluate a limit of a rational function as x tends to infinity?
Divide every term in the numerator and denominator by the highest power of \(x\) present. As \(x\to\infty\), any term of the form \(1/x^n\) tends to \(0\), leaving a simple ratio of the remaining coefficients.
What is the standard limit used for expressions like (1 + a/n)^n?
As \(n\to\infty\), \(\left(1+\tfrac{a}{n}\right)^n\) tends to \(e^{a}\). This standard result lets you evaluate such limits directly once you identify the value of \(a\).