Limits at Infinity

The central idea

A limit such as $\lim_{x\to\infty}f(x)=L$ describes the long-run output as $x$ grows without bound; $y=L$ is then a horizontal asymptote to the right. For rational functions, comparing numerator and denominator degrees determines the dominant powers, but end behavior at $+\infty$ and $-\infty$ may differ.

Definitions, hypotheses, and notation

The degree rules for rational functions are consequences of dominant-power division. Lower numerator degree gives limit zero; equal degrees give the leading-coefficient ratio; higher numerator degree produces unbounded or polynomial-like behavior rather than a horizontal asymptote. Long division can reveal a slant or higher-degree polynomial asymptote in the last case.

Radicals require special sign care. Factoring $x^2$ from a square root produces $|x|$, which equals $x$ as $x\to+\infty$ but $-x$ as $x\to-\infty$. This single distinction explains why $x/\sqrt{x^2+1}$ approaches $1$ on the right and $-1$ on the left.

Conceptual meaning

Far from the origin, lower-degree terms become negligible relative to the largest powers. Dividing by the dominant power makes this precise. A function may cross a horizontal asymptote; the asymptote describes behavior at infinity, not a barrier.

A dependable method and decision rule

1. Identify the dominant terms or divide numerator and denominator by the largest relevant power.
2. Use $1/x^k\to0$ as $x\to\pm\infty$ for $k>0$.
3. For radicals, factor the highest even power carefully and use $\sqrt{x^2}=|x|$.
4. Evaluate the positive and negative directions separately when signs may change.
5. Translate a finite end limit into the corresponding horizontal asymptote.

Fully worked example

Common mistakes and why they fail