Limits involving infinity
Limits Involving Infinity
Limits Involving Infinity is a topic in Limits and Continuity in the California Calculus Standards. It is aligned to Standard 1.0, which requires students to understand limits involving infinity — infinite limits at a point (vertical asymptotes) and limits at infinity (horizontal asymptotes).
Limits involving infinity come in two forms: a function that grows without bound near a point, \(\lim_{x\to a} f(x)=\pm\infty\) (a vertical asymptote), and a limit as \(x\) grows, \(\lim_{x\to\infty} f(x)=L\) (a horizontal asymptote).
Theory
Limits can involve infinity in two ways. An infinite limit (the function heads to \(\pm\infty\) at a vertical asymptote), and a limit at infinity (the value settles toward a horizontal asymptote as \(x\) grows). This page evaluates both using \(\dfrac{1}{x^{n}}\to 0\) and degree comparison.
An infinite limit describes a function that grows without bound near a point: \(\lim_{x\to a} f(x)=+\infty\) or \(-\infty\). The line \(x=a\) is a vertical asymptote. (This is a description of the behavior, not a finite value.)
A limit at infinity asks what happens as \(x\) grows: \(\lim_{x\to\infty} f(x)=L\) means the graph levels off toward the horizontal asymptote \(y=L\).
The single most useful fact is that a constant over a growing power vanishes:
The vanishing-power fact and the degree rule for \(\dfrac{p(x)}{q(x)}\) as \(x\to\infty\):
where \(a,b\) are the leading coefficients of \(p\) and \(q\).
How to evaluate a limit at infinity
- Compare degrees of numerator and denominator, or divide every term by the highest power of \(x\).
- Send each \(\dfrac{1}{x^{n}}\) term to \(0\).
- Read off the result: \(0\), a ratio of leading coefficients, or \(\pm\infty\).
Divide top and bottom by the highest power, \(x\).
| \(\dfrac{3x+1}{x-2}\) | \(=\) | \(\dfrac{3+\dfrac{1}{x}}{1-\dfrac{2}{x}}\) |
| \(\lim_{x\to\infty}\) | \(=\) | \(\dfrac{3+0}{1-0}=3\) |
Equal degrees, so the limit is the ratio of leading coefficients.
The denominator has the higher degree, so it grows faster.
| \(\dfrac{2x+5}{x^2+1}\) | \(=\) | \(\dfrac{\dfrac{2}{x}+\dfrac{5}{x^2}}{1+\dfrac{1}{x^2}}\) |
| \(\lim_{x\to\infty}\) | \(=\) | \(\dfrac{0}{1}=0\) |
Same degree on top and bottom — take the leading coefficients.
| \(\lim_{x\to\infty}\dfrac{x^2-1}{2x^2+3}\) | \(=\) | \(\dfrac{1}{2}\) |
As \(x\to 2^{+}\), \(x-2\) is a small positive number, so \(\dfrac{1}{x-2}\to +\infty\). As \(x\to 2^{-}\), \(x-2\) is small negative, so \(\dfrac{1}{x-2}\to -\infty\).
| \(\lim_{x\to 2^{+}}\dfrac{1}{x-2}\) | \(=\) | \(+\infty\) |
| \(\lim_{x\to 2^{-}}\dfrac{1}{x-2}\) | \(=\) | \(-\infty\) |
The two sides differ, so the two-sided limit does not exist (there is a vertical asymptote at \(x=2\)).
Common pitfalls
Frequently asked questions
What is the difference between an infinite limit and a limit at infinity?
An infinite limit is \(x\to a\) with \(f\to\pm\infty\) (a vertical asymptote). A limit at infinity is \(x\to\pm\infty\) with \(f\to L\) (a horizontal asymptote).
How do you find the limit of a rational function at infinity?
Compare degrees. If the bottom degree is larger the limit is \(0\); if the degrees are equal it is the ratio of leading coefficients; if the top is larger it is \(\pm\infty\).
What is the limit of 1 over x as x approaches infinity?
It is \(0\). More generally \(\dfrac{1}{x^{n}}\to 0\) for any \(n>0\) as \(x\) grows without bound.
Does a limit of infinity mean the limit exists?
No. \(+\infty\) and \(-\infty\) describe unbounded behavior; the limit does not exist as a finite number. If the two sides disagree in sign, the two-sided limit does not exist either.