Calculus I · Limits and Continuity · lesson

When a Limit Does Not Exist

Concept

Learning objectives

Recognize and explain the major reasons a limit fails to exist: unequal one-sided limits, unbounded behavior, oscillation, or missing domain on one side.

How a Limit Can Fail

Writing $\mathrm{DNE}$ is not a full explanation. A good solution identifies the behavior responsible.

Failure mode 1: the sides disagree

This is the jump behavior already seen:

\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x).

Failure mode 2: unbounded behavior

For

f(x)=\frac1x,

as $x\to0^+$ , the outputs become arbitrarily large and positive. As $x\to0^-$ , they become arbitrarily negative. We write

\lim_{x\to0^+}\frac1x=+\infty, \qquad \lim_{x\to0^-}\frac1x=-\infty.

The two-sided limit does not exist as a real number.

Failure mode 3: endless oscillation

For

f(x)=\sin\left(\frac1x\right),

the argument $1/x$ becomes arbitrarily large as $x\to0$ . The sine function cycles between $-1$ and $1$ infinitely often, so the outputs do not approach one number.

Two branches of sin(1/x) oscillating rapidly near zero. — **Read the graph**
The function $\sin(1/x)$ oscillates increasingly rapidly near zero and has no limit there.

Failure mode 4: the function exists only on one side

The function $f(x)=\sqrt{x}$ has no real values for $x<0$ . Therefore, the right-hand limit

\lim_{x\to0^+}\sqrt{x}=0

exists, but an ordinary two-sided real limit at $0$ is not available because there is no left-side domain near zero.

At an endpoint of a domain or interval, a one-sided limit is often exactly the correct question. Later, continuity at endpoints will also use one-sided limits.

Problem 1

When asked why a limit does not exist, use one of these sentence frames:

• "The left-hand limit is $A$ and the right-hand limit is $B$ , and $A\ne B$ ." • "The function is unbounded near the target input." • "The function oscillates without approaching one output." • "The function has no domain values on one required side of the target."

After the explanation

Use the section idea

Reading lens

What are nearby outputs doing as the input approaches the target from both sides?

Mental model

Imagine tightening a window around the target input and watching where all nearby outputs are forced to gather.

Decision

Read the left-hand and right-hand behavior separately first; combine them only after both sides approach the same output.

Common trap

The function value at the target can be missing or deliberately moved, so never substitute a plotted dot for evidence from both sides.

Check yourself

You understand the section when you can explain a limit from a graph, table, and sentence without confusing it with the function value.

Source & rights

Original instruction with traceable references.

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