Conceptual Bottlenecks Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

A common bottleneck is understanding why

\lim_{x\to a}f(x)

does not require

f(a)

to be defined. Illustrate with

f(x) = \frac{x^2-1}{x-1}

a=1

1
At $x=1$ : $f(1) = \frac{0}{0}$ , which is undefined. So $f(1)$ does not exist.
2
For $x\ne 1$ : $f(x) = \frac{(x-1)(x+1)}{x-1} = x+1$ .
3
$\lim_{x\to 1}f(x) = \lim_{x\to 1}(x+1) = 2$ . The limit exists and equals 2.
4
The limit describes the trend of $f$ as $x$ approaches 1, regardless of $f(1)$ .

Answer

\lim_{x\to 1}\frac{x^2-1}{x-1} = 2,\quad\text{even though } f(1) \text{ is undefined}

The conceptual bottleneck is the distinction between a limit (a trend) and a value (a point). Overcoming it requires accepting that 'approaching

a

' and 'being at

a

' are fundamentally different ideas.

About Conceptual Bottlenecks

Specific concepts or ideas whose misunderstanding blocks progress across a wide range of related mathematical topics.

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