Calculus I · Limits and Continuity · lesson

Limits at Infinity and Horizontal Asymptotes

Learn Limits at Infinity and Horizontal Asymptotes with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-styl

Course progress28 of 47 core pages28 of 47

Where this chapter fits

Chapter 4: Infinite behavior

Read vertical, horizontal, and slant asymptotes through sign analysis and dominant-term reasoning.

Reading lens: Is the function growing without bound near a finite input, or settling into end behavior as the input grows? Keep that question in view while reading Limits at Infinity and Horizontal Asymptotes; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Vertical Asymptotes and One-Sided Sign Analysis to Polynomial and Slant Asymptotes. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Interpret $x\to\pm\infty$ ; find horizontal asymptotes by comparing dominant terms; evaluate rational-function end behavior.

Limits at Infinity

The notation

\lim_{x\to\infty}f(x)=L

asks what happens to outputs as inputs become arbitrarily large and positive. The notation

\lim_{x\to-\infty}f(x)=L

asks what happens far to the left.

If either end limit equals a finite number $L$ , then $y=L$ is a horizontal asymptote on that end.

Concept

A horizontal asymptote describes the far-away behavior of a graph. The graph may cross it nearby. An asymptote is not an electric fence. It is a long-term trend.

Reciprocal powers

For every positive integer $n$ ,

\lim_{x\to\infty}\frac1{x^n}=0, \qquad \lim_{x\to-\infty}\frac1{x^n}=0.

These facts drive the degree rules for rational functions.

Guided walkthrough

One over a growing number

Evaluate

\lim_{x\to\infty}\frac1x.

Show worked solution

As $x$ becomes $10,100,1000,\ldots$ , the values $1/x$ become

0.1,0.01,0.001,\ldots

which approach zero. Therefore,

\boxed{0}.

Equal degrees

Worked example

Divide by the largest denominator power

Evaluate

\lim_{x\to\infty}\frac{3x^2-2x+5}{x^2+4}.

Show worked solution

Divide every term by $x^2$ , the largest power in the denominator:

\frac{3x^2-2x+5}{x^2+4} =\frac{3-\frac2x+\frac5{x^2}}{1+\frac4{x^2}}.

As $x\to\infty$ , all reciprocal terms approach zero:

\frac{3-0+0}{1+0}=\boxed{3}.

Thus $y=3$ is a horizontal asymptote.

Interactive checkend-degree-01

Evaluate $\lim_{x\to\infty}\frac{3x^2+1}{2x^2-5}$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Equal degrees: use the leading coefficients.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Numerator degree smaller than denominator degree

Worked example

The denominator wins

Evaluate

\lim_{x\to-\infty}\frac{5x-1}{x^2+3}.

Show worked solution

Divide by $x^2$ :

\frac{\frac5x-\frac1{x^2}}{1+\frac3{x^2}}.

Every reciprocal term approaches zero, so

\boxed{0}.

The quadratic denominator grows faster in magnitude than the linear numerator.

Numerator degree larger than denominator degree

If the numerator degree is larger, there is no finite horizontal asymptote. The quotient may grow like a polynomial.

Worked example

Unbounded end behavior

Evaluate

\lim_{x\to\infty}\frac{2x^3-x}{x^2+1}.

Show worked solution

The leading behavior is approximately

\frac{2x^3}{x^2}=2x,

which grows to $+\infty$ . More formally, divide by $x^2$ :

\frac{2x-\frac1x}{1+\frac1{x^2}}.

The numerator grows without bound while the denominator approaches $1$ , so

\boxed{+\infty}.

Theorem

Rational-Function Degree Rules

For $f(x)=p(x)/q(x)$ :

• If $\deg p<\deg q$ , then $f(x)\to0$ as $x\to\pm\infty$ . • If $\deg p=\deg q$ , the limit is the ratio of leading coefficients. • If $\deg p>\deg q$ , there is no finite horizontal asymptote; use division or dominant terms to determine the end behavior.

Graph reading guide

The rational function approaches its horizontal asymptote $y=3$ at both ends.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Calculus I · Limits and Continuity · lesson

Limits at Infinity and Horizontal Asymptotes

Learn Limits at Infinity and Horizontal Asymptotes with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-styl

Course progress28 of 47 core pages28 of 47

Where this chapter fits

Chapter 4: Infinite behavior

Read vertical, horizontal, and slant asymptotes through sign analysis and dominant-term reasoning.

Concept

Learning objectives

Interpret $x\to\pm\infty$ ; find horizontal asymptotes by comparing dominant terms; evaluate rational-function end behavior.

Limits at Infinity

The notation

\lim_{x\to\infty}f(x)=L

asks what happens to outputs as inputs become arbitrarily large and positive. The notation

\lim_{x\to-\infty}f(x)=L

asks what happens far to the left.

If either end limit equals a finite number $L$ , then $y=L$ is a horizontal asymptote on that end.

Concept

A horizontal asymptote describes the far-away behavior of a graph. The graph may cross it nearby. An asymptote is not an electric fence. It is a long-term trend.

Reciprocal powers

For every positive integer $n$ ,

\lim_{x\to\infty}\frac1{x^n}=0, \qquad \lim_{x\to-\infty}\frac1{x^n}=0.

These facts drive the degree rules for rational functions.

Guided walkthrough

One over a growing number

Evaluate

\lim_{x\to\infty}\frac1x.

Show worked solution

As $x$ becomes $10,100,1000,\ldots$ , the values $1/x$ become

0.1,0.01,0.001,\ldots

which approach zero. Therefore,

\boxed{0}.

Equal degrees

Worked example

Divide by the largest denominator power

Evaluate

\lim_{x\to\infty}\frac{3x^2-2x+5}{x^2+4}.

Show worked solution

Divide every term by $x^2$ , the largest power in the denominator:

\frac{3x^2-2x+5}{x^2+4} =\frac{3-\frac2x+\frac5{x^2}}{1+\frac4{x^2}}.

As $x\to\infty$ , all reciprocal terms approach zero:

\frac{3-0+0}{1+0}=\boxed{3}.

Thus $y=3$ is a horizontal asymptote.

Interactive checkend-degree-01

Evaluate $\lim_{x\to\infty}\frac{3x^2+1}{2x^2-5}$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Equal degrees: use the leading coefficients.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Numerator degree smaller than denominator degree

Worked example

The denominator wins

Evaluate

\lim_{x\to-\infty}\frac{5x-1}{x^2+3}.

Show worked solution

Divide by $x^2$ :

\frac{\frac5x-\frac1{x^2}}{1+\frac3{x^2}}.

Every reciprocal term approaches zero, so

\boxed{0}.

The quadratic denominator grows faster in magnitude than the linear numerator.

Numerator degree larger than denominator degree

If the numerator degree is larger, there is no finite horizontal asymptote. The quotient may grow like a polynomial.

Worked example

Unbounded end behavior

Evaluate

\lim_{x\to\infty}\frac{2x^3-x}{x^2+1}.

Show worked solution

The leading behavior is approximately

\frac{2x^3}{x^2}=2x,

which grows to $+\infty$ . More formally, divide by $x^2$ :

\frac{2x-\frac1x}{1+\frac1{x^2}}.

The numerator grows without bound while the denominator approaches $1$ , so

\boxed{+\infty}.

Theorem

Rational-Function Degree Rules

For $f(x)=p(x)/q(x)$ :

Graph reading guide

The rational function approaches its horizontal asymptote $y=3$ at both ends.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Chapter 4: Infinite behavior

Limits at Infinity

Reciprocal powers

Equal degrees

Numerator degree smaller than denominator degree

Numerator degree larger than denominator degree

Related resources

Original instruction with traceable references.

Chapter 4: Infinite behavior

Limits at Infinity

Reciprocal powers

Equal degrees

Numerator degree smaller than denominator degree

Numerator degree larger than denominator degree

Related resources

Original instruction with traceable references.