Calculus I · Limits and Continuity · lesson

Radical Limits at Infinity

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}.

After the explanation

Use the section idea

Reading lens

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

Source & rights

Original instruction with traceable references.

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