Calculus I · Limits and Continuity · lesson

Limits at Holes and Undefined Points

A function can have a limit where it is undefined

Consider

f(x)=\frac{x^2-4}{x-2}.

At $x=2$ , the formula is undefined. For $x\ne2$ , however,

f(x)=\frac{(x-2)(x+2)}{x-2}=x+2.

So the graph is the line $y=x+2$ with one point removed.

Reference table
$x$	$f(x)$
1.9	3.9
1.99	3.99
1.999	3.999
2.001	4.001
2.01	4.01
2.1	4.1

Both sides approach $4$ , so

\boxed{\lim_{x\to2}\frac{x^2-4}{x-2}=4}.

Graph of y equals x plus 2 with a removable hole at (2, 4). — **Read the graph**
The function is undefined at $x=2$ , but nearby outputs approach $4$ .

Method

To estimate a limit from a table, use inputs on both sides of the target. The table should move progressively closer to the target. A table gives evidence, not proof; algebra or a trustworthy graph is usually needed to confirm the pattern.

Worked example

Table, graph, and algebra agree

Estimate and then evaluate

\lim_{x\to1}\frac{x^3-1}{x-1}.

Show worked solution

Numerical evidence.

Reference table
$x$	$\dfrac{x^3-1}{x-1}$
$0.9$	$2.71$
$0.99$	$2.9701$
$0.999$	$2.997001$
$1.001$	$3.003001$
$1.01$	$3.0301$
$1.1$	$3.31$

The values appear to approach $3$ .

Algebraic confirmation. Use the difference-of-cubes identity:

x^3-1=(x-1)(x^2+x+1).

For $x\ne1$ ,

\frac{x^3-1}{x-1}=x^2+x+1.

Therefore,

\begin{aligned} \lim_{x\to1}\frac{x^3-1}{x-1} &=\lim_{x\to1}(x^2+x+1)\\ &=1^2+1+1\\ &=\boxed{3}. \end{aligned}

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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