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

Graphical or geometric meaning

A graphing window must extend far enough to show end behavior; a local bend can be misleading. The dominant-term ratio gives the height the curve approaches while the signs of small remaining terms indicate from which side it approaches.

Common mistakes and why they fail

Verification and reasonableness checks

• Estimate at a very large positive or negative input.
• Compare polynomial degrees before doing algebra.
• Confirm the claimed horizontal asymptote matches a finite limit.

Compare growth before doing algebra

For rational functions, divide by the highest denominator power and compare degrees. Lower numerator degree yields zero, equal degrees yield the ratio of leading coefficients, and higher numerator degree produces polynomial-like behavior. The sign as $x\to-\infty$ still depends on parity and leading coefficients, so handle the two directions separately. For nonrational expressions, the growth hierarchy is useful: exponentials eventually dominate powers, and powers dominate logarithms. A horizontal asymptote describes only end behavior and may be crossed at finite inputs. Verify a result by checking that the proposed leading term has the same scale and sign as the original expression for large positive or negative inputs.

Practice

1. Find $\lim_{x\to\infty}(2x+1)/(x-4)$.
2. Find $\lim_{x\to\infty}1/x^3$.
3. Find $\lim_{x\to-\infty}x/\sqrt{x^2+1}$.

Connections and next steps

• Infinite Limits
• Limits
• Curve Sketching