Why does the epsilon-delta definition require x to not equal a?

Q: Why does the epsilon-delta definition require x to not equal a?

The definition requires $x \neq a$ (expressed as $0 < |x - a|$) because the limit describes the behavior of the function *as it approaches* the point $a$, not its value *at* the point $a$. The function might be undefined at $x = a$, or its value $f(a)$ might differ from the limit $L$. Excluding $x = a$ ensures the limit depends only on the surrounding values.

The definition requires

x \neq a

(expressed as

0 < |x - a|

) because the limit describes the behavior of the function *as it approaches* the point

a

, not its value *at* the point

a

. The function might be undefined at

x = a

, or its value

f(a)

might differ from the limit

L

. Excluding

x = a

ensures the limit depends only on the surrounding values.

Conditions

The limit is concerned with the trend of $f(x)$ near $a$ .
$f(a)$ may be undefined or discontinuous at $a$ .

Reasoning, step by step

Recognize that the limit $L$ is the value $f(x)$ approaches as $x \to a$ .
Understand that the actual value $f(a)$ is irrelevant to the limit's existence or value.
Apply the condition $0 < |x - a|$ to restrict the domain to a punctured neighborhood of $a$ .
Conclude that this exclusion allows the limit to exist even if the function has a hole or jump at $a$ .

Example

The speaker explicitly states, 'the one thing I can't guarantee you is what happens when x is equal to a.' The board shows the condition $0<|x-a|<\delta$ , which mathematically enforces $x \neq a$ .

Common misconceptions

Believing that the limit depends on the function's value exactly at $x = a$ .
Thinking that a function must be defined at $a$ for the limit to exist.
Confusing the limit of a function with its continuity at a point.

Watch the explanation

YouTubeEpsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy

9:10 – 9:20Watch this moment ↗

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