AcademyFoundations And Limits
Academy
Asymptotic Behaviour
Level 1 - Calculus topic page in Foundations And Limits.
Principle
Asymptotic behaviour describes how a function behaves near excluded points or far from the origin. Asymptotes are guide lines approached by the graph.
Notation
\(x=a\)
vertical asymptote candidate
\(y=L\)
horizontal asymptote candidate
\(x\to\pm\infty\)
input grows without bound
The Core Method
Vertical asymptotes come from infinite limits near finite input values. Horizontal asymptotes come from limits as \(x\to\infty\) or \(x\to-\infty\).
Horizontal asymptote
\[\lim_{x\to\infty}f(x)=L\Longrightarrow y=L\]
For rational functions, compare the highest powers of numerator and denominator to decide end behaviour.
Worked Cases
Question
Find the horizontal asymptote of \(f(x)=\frac{3x^2+1}{2x^2-5}\).
Answer
The numerator and denominator have the same highest power, \(x^2\). Divide by \(x^2\): \(f(x)=\frac{3+1/x^2}{2-5/x^2}\). As \(x\to\pm\infty\), the fractions \(1/x^2\) and \(5/x^2\) go to zero. The limit is \(3/2\), so the horizontal asymptote is \(y=3/2\).
Examples
Question
Find the vertical asymptote of \(f(x)=\frac{x+1}{x-4}\).
Answer
The denominator is zero at \(x=4\), while the numerator is \(5\), not zero. Therefore \(f(x)\) has an infinite limit near \(4\), and the vertical asymptote is \(x=4\).