CHANGE AND LIMITS 05
Limits at Infinity
Limits at infinity describe the long-run behaviour of a function as the input becomes very large positive or negative.
The main idea
The notation
\[\boxed{\lim_{x\to\infty}f(x)=L}\]means that \(f(x)\) approaches \(L\) as \(x\) increases without bound.
1. A simple example
For \(f(x)=1/x\),
\[\lim_{x\to\infty}\frac1x=0.\]The function gets closer to the \(x\)-axis as \(x\) grows.
2. Horizontal asymptotes
If \(f(x)\to L\) as \(x\to\infty\) or \(x\to-\infty\), then \(y=L\) is a horizontal asymptote in that direction.
3. Rational functions with equal degrees
For
\[f(x)=\frac{2x+1}{x+1},\]divide numerator and denominator by \(x\):
\[\frac{2+1/x}{1+1/x}\to2.\]Therefore \(\lim_{x\to\infty}f(x)=2\).
4. Comparing polynomial degrees
For rational functions, the highest powers control long-run behaviour. If the denominator degree is larger, the limit is often \(0\). If the degrees are equal, the limit is the ratio of leading coefficients.
equal degrees → ratio of leading coefficients
numerator degree larger → inspect growth
5. Positive and negative infinity can differ
Long-run behaviour should be checked in both directions when relevant:
\[\lim_{x\to\infty}f(x),\qquad\lim_{x\to-\infty}f(x).\]Some functions have the same end behaviour; others do not.
Limits at infinity describe end behaviour rather than behaviour near a finite point. They are closely connected to horizontal asymptotes and can often be evaluated by comparing dominant terms.