Calculus I · Limits and Continuity · lesson

Limits With Absolute Values and Piecewise Functions

Concept

Learning objectives

Rewrite absolute values on each side of a target and evaluate one-sided limits before deciding whether a two-sided limit exists.

Absolute Values and Piecewise Rules

The absolute value is piecewise:

|u|= \begin{cases} u,&u\ge0,\\ -u,&u<0. \end{cases}

The sign of the inside expression determines which rule applies.

Guided walkthrough

$|x|/x$

Evaluate

\lim_{x\to0}\frac{|x|}{x}.

Show worked solution

For $x>0$ , $|x|=x$ , so

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

Therefore,

\lim_{x\to0^+}\frac{|x|}{x}=1.

For $x<0$ , $|x|=-x$ , so

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

Therefore,

\lim_{x\to0^-}\frac{|x|}{x}=-1.

The sides disagree, so

\boxed{\lim_{x\to0}\frac{|x|}{x}=\mathrm{DNE}}.

Worked example

A shifted absolute value

Evaluate

\lim_{x\to3}\frac{|x-3|}{x-3}.

Show worked solution

When $x>3$ , the inside $x-3$ is positive, so

|x-3|=x-3

and the quotient equals $1$ .

When $x<3$ , the inside $x-3$ is negative, so

|x-3|=-(x-3)

and the quotient equals $-1$ .

Thus,

\lim_{x\to3^-}\frac{|x-3|}{x-3}=-1,

\lim_{x\to3^+}\frac{|x-3|}{x-3}=1.

Therefore the two-sided limit does not exist.

Worked example

An absolute value whose limit does exist

Evaluate

\lim_{x\to2}\frac{|x-2|}{\sqrt{|x-2|}}.

Show worked solution

For $x\ne2$ , $|x-2|>0$ , and

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

As $x\to2$ , the distance $|x-2|\to0$ . Therefore,

\sqrt{|x-2|}\to0,

\boxed{0}.

Absolute value does not automatically cause a limit to fail. It causes one-sided analysis when the formula changes sign or form across the target.

After the explanation

Use the section idea

Reading lens

What did direct substitution reveal, and which algebraic move removes the obstacle without changing nearby behavior?

Mental model

Substitution is a diagnostic first move: a real number usually finishes the problem, while an indeterminate form asks for a structural rewrite.

Decision

Match the obstacle to the algebra—factor polynomial zeros, rationalize radicals, combine complex fractions, and split absolute values into one-sided cases.

Common trap

Zero over zero is not an answer, and cancellation is legal only for factors after the expression has been rewritten as a product.

Check yourself

You are ready to move on when you can justify why each rewrite preserves nearby values even if the original expression is undefined at the target.

Source & rights

Original instruction with traceable references.

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