Limits

Intuitively (for the formal definition, see Epsilon-Delta Construction of the Limit), a limit of function $𝑓 (𝑥)$ as $𝑥$ approaches $0$

l i m 𝑥 \to 0 𝑓 (𝑥)

is what $𝑓 (𝑥)$ gets close to as $𝑥$ gets close to $0$ . In some cases this is obvious — for example, for any function Continuous and defined at $𝑓 (𝑎)$ ,

l i m 𝑥 \to 𝑎 𝑓 (𝑥) = 𝑓 (𝑥)

In other cases, it may be a value which the function visibly approaches

Limits can be calculated using Limit Laws. They are generalised by the category-theoretic notion of Limits and colimits

One-sided limits

Limits can be defined as approaching from the left hand side ( $-$ ), or as approaching from the right hand side ( $+$ ). These are written as $l i m 𝑥 \to 𝐿 -$ and $l i m 𝑥 \to 𝐿 +$ respectively. Hence,

l i m 𝑥 \to 𝑎 𝑓 (𝑥) = 𝐿 ⟺ l i m 𝑥 \to 𝑎 + 𝑓 (𝑥) = 𝐿 \land l i m 𝑥 \to 𝑎 - 𝑓 (𝑥) = 𝐿

Undefined limits

Limits Do Not Exist (DNE) in cases where

For a bilateral limit, $𝑥 \to 𝑎 +$ and $𝑥 \to 𝑎 -$ give different results (see above).
The limit is $\pm \infty$ .
The input limit (i.e. what $𝑥$ approaches) is a finite distance outside the domain of $𝑓 (𝑥)$ .

Multivariable limits

See Multivariable limits.

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