Calculus I · Limits and Continuity · lesson

Radical Limits at Infinity

Learn Radical Limits at Infinity with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.

Course progress30 of 47 core pages30 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 Radical Limits at Infinity; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Polynomial and Slant Asymptotes to Infinity Minus Infinity Limits. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Concept

Learning objectives

Factor the highest power from a radical correctly; use $\sqrt{x^2}=|x|$ ; evaluate radical differences by rationalization.

Radicals at Infinity and the Absolute-Value Trap

The identity

\boxed{\sqrt{x^2}=|x|}

is essential. At positive infinity, $|x|=x$ . At negative infinity, $|x|=-x$ .

Guided walkthrough

Why the sign changes

Evaluate

\lim_{x\to-\infty}\frac{\sqrt{x^2}}{x}.

Show worked solution

Since $\sqrt{x^2}=|x|$ ,

\frac{\sqrt{x^2}}{x}=\frac{|x|}{x}.

For negative $x$ , $|x|=-x$ . Therefore,

\frac{|x|}{x}=\frac{-x}{x}=-1.

Hence

\boxed{-1}.

Writing $\sqrt{x^2}=x$ would incorrectly give $+1$ .

Worked example

The same radical on two ends

Evaluate

\lim_{x\to\infty}\frac{\sqrt{x^2+1}}{x} \qquad\text{and}\qquad \lim_{x\to-\infty}\frac{\sqrt{x^2+1}}{x}.

Show worked solution

Factor $x^2$ inside the radical:

\sqrt{x^2+1}=\sqrt{x^2\left(1+\frac1{x^2}\right)} =|x|\sqrt{1+\frac1{x^2}}.

At positive infinity, $|x|/x=1$ , so

\boxed{1}.

At negative infinity, $|x|/x=-1$ , so

\boxed{-1}.

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.

Chapter 4: Infinite behavior

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

Radicals at Infinity and the Absolute-Value Trap

The identity

\boxed{\sqrt{x^2}=|x|}

is essential. At positive infinity,

|x|=x

. At negative infinity,

|x|=-x

Guided walkthrough

Why the sign changes

Evaluate

\lim_{x\to-\infty}\frac{\sqrt{x^2}}{x}.

Show worked solution

Since $\sqrt{x^2}=|x|$ ,

\frac{\sqrt{x^2}}{x}=\frac{|x|}{x}.

For negative $x$ , $|x|=-x$ . Therefore,

\frac{|x|}{x}=\frac{-x}{x}=-1.

Hence

\boxed{-1}.

Writing $\sqrt{x^2}=x$ would incorrectly give $+1$ .

Worked example

The same radical on two ends

Evaluate

\lim_{x\to\infty}\frac{\sqrt{x^2+1}}{x} \qquad\text{and}\qquad \lim_{x\to-\infty}\frac{\sqrt{x^2+1}}{x}.

Show worked solution

Factor $x^2$ inside the radical:

\sqrt{x^2+1}=\sqrt{x^2\left(1+\frac1{x^2}\right)} =|x|\sqrt{1+\frac1{x^2}}.

At positive infinity, $|x|/x=1$ , so

\boxed{1}.

At negative infinity, $|x|/x=-1$ , so

\boxed{-1}.

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

Radicals at Infinity and the Absolute-Value Trap

Related resources

Original instruction with traceable references.

Chapter 4: Infinite behavior

Radicals at Infinity and the Absolute-Value Trap

Related resources

Original instruction with traceable references.